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NEWCOMB'S MATHEMATICAL COURSE
ELEMENTS
DIFFERENTIAL AND INTEGRAL
QALCULUS
' - n
'I' Ok:,/
BY ' '' t. h h Y
siko^ :n;ewoomb
Professor of MathemaUcs in the Johns Hopkins University
NEW YORK
HENRY HOLT AND COMPANY
1887
CoPYiaoHT, 1887,
BY
HENRY HOLT & CO.
Ul
PREFACE.
The present work is intended to contain about as much
of the Calcuhis as an undergraduate student, either in Arts
or Science, can be expected to master during his regular
course. He may find more exercises than he has time to
work out; in this case it is suggested that he only work
enough to show that he understands the principles they are
designed to elucidate.
The most difficult question which arises in treating the
subject is how the first principles should be presented to the
mind of the beginner. The author has deemed it best to be-
gin by laying down the logical basis on which the whole
superstructure must ultimately rest. It is now well under-
stood that the method of limits forms the only rigorous basis
for the infinitesimal calculus, and that infinitesimals can be
used with logical rigor only when based on this method, that
is, when considered as quantities approaching zero as their
limit. When thus defined, no logical difficulty arises in their
use; they fiow naturally from the conception of limits, and
they are therefore introduced at an early stage in the present
work.
The fundamental principles on which the use of infinitesi-
mals is based are laid down in the second chapter. But it is
not to be expected that a beginner will fully grasp these prin-
ciples until he has become familiar with the mechanical pro-
cess of differentiation, and with the application of the calcu-
iv PREFACM.
Ins to special problems. It may therefore be found best to
begin with a single careful reading of the chapter, and after-
ward to use it for reference as the student finds occasion to
apply the principles laid down in it.
The author is indebted to several friends for advice and
assistance in the final revision of the work. Professor John
E. Clark of the Sheffield Scientific School and Dr. Fabian
Franklin of the Johns Hopkins University supplied sugges-
tions and criticisms which proved very helpful in putting the
first three chapters into shape. Miss E. P. Brown of Wash-
ington has read all the proofs, solving most of the problems as
she went along in order to test their suitability.
CONTENTS.
PAET I.
TEE DIFFEBENTIAL OALOULUS.
PAGE
Chaptek I. Op "Vabiablbs and Functions 3
§1. Nature of Functions. 3. Their Classification. 3. Func-
tional Notation. 4. Functions of Several Variables. 5. Func-
tions of Functions. 6. Product of the First n Numbers. 7. Bi-
nomial CoeflBcients. 8. Graphic Representation of Functions.
9. Continuity and Discontinuity of Functions. 10. Many- valued
Functions.
Chaptee II. Of Limits and Infinitesimals 17
§ 11. Limits. 12. Infinites and Infinitesimals. 13. Properties.
14. Orders of Infinitesimals. 15. Orders of Infinites.
Chaptek III. Of Differentials and Dekivatives 35
§ 16. Increments of Variables. 17. First Idea of DifEerentials
and Derivatives. 18. Illustrations. 19. Illustration by Velocities.
20. Geometrical Illustration.
Chapter IV. Differentiation of Explicit Functions 31
§ 21. The Process of Differentiation in General. 32. DifEeren-
tials of Sums. 33. Differential of a Multiple. 34. Differential of
a Constant. 35. Differentials of Products and Powers. 36. Dif-
ferential of a Quotient of Two Variables. 37. Differentials of Ir-
rational Expressions. 38. Logarithmic Functions. 39. Expo-
nential Functions. 30. The Trigonometric Functions. 31. Cir-
cular Functions. 32. Logarithmic Differentiation. 33. Velocity
or Derivative with Respect to the Time.
Chapter V. Functions of Several Variables and Implit
cit Functions 54
§ 34. Partial Differentials and Derivatives. 35. Total Differen-
tials. 36. Principles Involved in Partial Differentiation. 37. Dif-
Vi CONTENTS.
PASE
fcrentiation of Implicit Functions. 38. Implicit Functions of Sev-
eral Variables. 39. Case of Implicit Functions expressed by
Simultaneous Equations. 40. Functions of Functions. 41. Func-
tions of Variables, some of which are Functions of the Others.
43. Extension of the Principle. 43. Nomenclature of Partial
Derivatives. 44. Dependence of the Derivative upon the Form
of the Function.
Chapter VI. DBRivATrvES of EiaHEB Orders 74
§45. Second Derivatives. 46. Derivatives of Any Order.
47. Special Forms of Derivatives of Circular and Exponential
Functions. 48. Successive Derivatives of an Implicit Function.
49. Successive Derivatives of a Product. 50. Successive Deriva-
tives with Respect to Several Equicrescent Variables. 51. Result
of Successive Differentiations independent of the Order of the
Difflerentiations. 53. Kotation for Powers of a Differential or
Derivative.
Chapter VII. Speciai Cases of Successivb Dbrivatites. . . 86
§ 58. Successive Derivatives of a Power of a Derivative. 54. De-
rivatives of Functions of Functions. 55. Change of the Equicres-
cent Variable. 56. Two Variables connected by a Third.
Chapter VIII. DBVBiiOPMENTS ts Series 95
§ 57. Classification of Series. 58. Convergence and Divergence
of Series. 59. Maclaurin's Theorem. 60. Ratio of the Circum-
ference of a Circle to its Diameter. 61. Use of Symbolic Nota-
tion for Derivatives^ 63. Taylor's Theorem. 63. Identity of
Taylor's and Maclaurin's Theorems. 64. Cases of Failure of
Taylor's and Maclaurin's Theorems. 65. Extension of Taylor's
Theorem to Functions of Several Variables. 66. Hyperbolic
Functions.
Chapter IX. Maxima aud Minima of Functions of a Sin-
gle Variable 117
§67. Definition of Maximum Value and Minimum Value.
68. Method of finding Maximum and Minimum Values of a Func-
tion. 69. Case when the Function which is to be a Maximum or
Minimum is expressed as a Function of Two or More Variables
connected by Equations of Condition.
Chapter X. Indbtbrminatb Forms 128
§ 70. Examples of Indeterminate Forms. 71. Evaluation of
the Form -r-. 73. Forms — and X oo . 73. Form oo — oo .
oo
75. Forms 0° and co ".
CONTENTS. VU
PASS
Chapter XI. Of Platte Oubtbs 137
§ 76. Forms of the Equations of Curves. 77. Inflnitesimal Ele-
ments of Curves. 78. Properties of Infinitesimal Ares and
Chords. 79. Expressions for Elements of Curves. 80. Equa-
tions of Certain Noteworthy Curves. The Cycloid. 81. The
Lemniscate. 83. The Archimedean Spiral. 83. The Logarith-
mic Spiral.
Chaptbb XII. Tangents and Nokmais 147
§84 Tangent and Normal compared with Subtangent and
Subnormal. 85. General Equation for a Tangent. 86. Sub-
tangent and Subnormal. 87. Modified Forms of the Equation.
88. Tangents and Normals to the Conic Sections. 89. Length of
the Perpendicular from the Origin upon a Tangent or Noi-mal.
90. Tangent and Normal in Polar Co-ordinates. 91. Perpendicular
from the Pole upon the Tangent or Normal. 93. Equation of
Tangent and Normal derived from Polar Equation of the Curve.
Chapter XIII. Of Asymptotes, Singulab Pointb and
Clkve-teacing 157
§93. Asymptotes. 94. Examples of Asymptotes. 95. Points
of Inflection. 96. Singular Points of Curves. 97. Condition of
Singular Points. 98. Examples of Double-points. 99. Curve-
tracing.
Chaptbb XIV. Thbokt of Envelopes 169
§100. Envelope of a Family of Lines. 101. All Lines of a
Family tangent to the Envelope. 103. Examples and Applications.
Chapter XV. Of Cuetatube, Etoltjtes and Involutes 180
§103. Position; Direction; Curvature. 104. Contacts of Differ-
ent Orders. 105. Intersection or Non-intersection of Curves ac-
cording to the Order of Contact. 106. Radius of Curvature.
107. The Osculating Circle. 108. Radius of Curvature when the
Abscissa is not taken as the Independent Variable. 109. Ra-
dius of Curvature of a Curve referred to Polar Co-ordinates.
110. Evolutes and Involutes. 111. Case of an Auxiliary Variable.
113. The Evolute of the Parabola. 113. Evolute of the Ellipse.
114. Evolute of the Cycloid. 115. Fundamental Properties of
the Evolute. 116. Involutes.
viu CONTENTS.
PAET II.
THE INTEGRAL CAL0ULU8.
PAOK
Chapteb I. The Elbmbntaby Forms of Intbgkation 201
§117. Deflnition of Integration. 118. Arbitrary Constant of
Integration. 119. Integration of Entire Functions. 120. The
Logarithmic Function. 131. Another Method of obtaining the
Logarithmic Integral. 123. Exponential Functions. 123. The
Elementary Forms of Integration.
Chapter II. Integrals immbdiatelt reducible to the
Elementary Forms 209
§ 124. Integrals reducible to the Form / ydy. 125. Appli-
cation to the Case of a Falling Body. 136. Reduction to the Loga-
rithmic Form. 127. Trigonometric Forms. 138. Integration of
a^ "^^ ¥^r 129- Integrals of the Form f^~^.
130. Inverse Sines and Cosines as Integrals. 131. Two Forms of
Integrals expressed by Circular Functions. 133. Integration of
133. Integration of . 134. Exponen-
tial Forms.
Chapter III. Integration by Rational Transformations. . 333
§ 135. Integration of ^ ' —dx, - — j-t-t- and
a!» ' (a + Ja;)" a + fee ± «''
136. Reduction of Rational Fractions in general. 137. Integra-
tion by Parts.
Chapter IV. Integration of Irrational Algebraic Dif-
ferentials 333
§138. When Fractional Powers of the Independent Variable
enter into the Expression. 139. Cases when the Given Dififeren-
tial Contains an Irrational Quantity of the Form 4/a + fes + esfl.
cLt
140. Integration of dB = — 141. General Theory
r V (zr* -}-br — 1
of Irrational Binomial Differentials. 143. Special Cases when
TO + 1 = ra, OT m + l + np = — n. 143. Forms of Reduction
of Irrational Binomials. 144. Formulae A and B, in which m is
increased or diminished by n. 145. Ponnulae C and D, in which
p is increased or diminished by 1. 146. Effect of the Formulae.
147. Case of Failure in this Reduction.
CONTENTS. ix
PAGE
Chaptek v. Integration of Tkanscendbnt Ftihctions 246
§ 148. Integration of / &^ cos nxdx and / Cmx sin vaxla!.
149. Integration of sin" » cos" xd/x. 150. Special Cases of / sin"" x
co^xdx. 151. Integration of „ . „ — , — = 5-. 152. Inteara-
m^ sin^ x-\-n? cos-' x "
tion of — —^ . 153. Special Cases of the Last Two Forms.
a -\- cos y
154. Integration of sin mx cos nxdx. 155. Integration by Devel-
opment in Series.
Chaptbk VI. Op Definite Integrals 255
§156. Successive Increments of a Variable. 157. Differential
of an Area. 158. The Formation of a Definite Integral. 159. Two
Conceptions of a Definite Integral. 160. Differentiation of a
Definite Integral with respect to its Limits. 161. Examples and
Exercises in finding Definite Integrals. 162. Failure of the
Method when the Function becomes Infinite. 163. Change of
Variable in Definite Integrals. 164. Subdivision of a Definite In-
tegral. 165. Definite Integrals through Integration by Parts.
Chapter VII. Successive Intesbation 372
§ 166. DLfferentiation under the Sign of Integration. 167. Ap-
plication of the Principle to Definite Integrals. 168. Integration
by means of Differentiating Known Integrals. 169. Application
to a Special Case. 170. Double Integrals. 171. Value of a Func-
tion of Two Variables obtained from its Second Derivative.
172. Triple and Multiple Integrals. 173. Definite Double Inte-
grals. 174. Definite Triple and Multiple Integrals. 175. Product of
e-x'dx.
Chapter VIII. Ebctification and Quadrature 285
§177. The Eectification of Curves. 178. The Parabola. 179. The
Ellipse. 180. The Cycloid. 181. The Archimedean Spiral.
183. The Logarithmic Spiral. 183. The Quadrature of Plane
Figures. 184. The Parabola. 185. The Circle and the Ellipse.
186. The Hyperbola. 187. The Lemniscate. 188. The Cycloid.
Chapter IX. The Cubatubb of Volumes 297
§189. General Formulae. 190. The Sphere. 191. The Pyra-
mid. 192. The Ellipsoid. 193. Volume of any Solid of Revolu-
tion. 194. The Paraboloid of Revolution. 195. The Volume gen-
erated by the Revolution of a Cycloid around its Base. 196. The
Hyperboloid of Revolution of Two Nappes. 197. Ring-shaped
Solids of Revolution. 198. Application to the Circular Ring.
199. Quadrature of Surfaces of Revolution. 200. Examples of
Surfaces of Revolution.
PART I.
THE DIFFERENTIAL CALCULUS.
USE OP THE SYMBOL =.
The symbol = of identity as employed in this work indi-
cates that the single letter on one side of it is used to repre-
sent the expression or thing defined on the other side of it.
When the single letter precedes the symbol =, the latter
may commonly be read is put for, or is defined as.
When the single letter follows the symbol, the latter may
be read which let us call.
In each case the equality of the quantities on each side of
H does not follow from anything that precedes, bat is assumed
at the moment. But haying once made this assumption, any
equations which may flow from it are expressed by the sign
=, as usual.
PART I.
THE DIFFERENTIAL CALCULUS.
CHAPTER I.
OF VARIABLES AND FUNCTIONS.
1. In the higher mathematics we conceive ourselves to be
dealing with pairs of quantities so related that the value of
one depends upon that of the other. For each value which
we assign to one we conceive that there is a corresponding
value of the other.
For example, the time required to perform a journey is a
function of the distance to be travelled, because, other things
being equal, the time varies when the distance varies.
We study the relation between two such quantities by as-
signing values at pleasure to one, and ascertaining and com-
paring the corresponding values of the other.
The quantity to which we assign values at pleasure is called
the independent variable.
The quantity whose values depend upon those of the inde-
pendent variable is called a function of that variable.
Example I. If a train travels at the rate of 30 miles an
hour, and if we ask how long it will take the train to travel
15 miles, 30 miles, 60 miles, 900 miles, etc., we shall have for
the corresponding times, or functions of the distances, half an
hour, one hour, two hours, thirty hours, etc.
4 THE DIFFERENTIAL CALCULUS.
In thinking thus we consider the distance to be travelled as
the independent variable, and the time as the function of the
distance.
Example II. If between the quantities x and y we have
the equation
y = 3fla;',
we may suppose
a; = -l, 0, +1, +2, +3yetc.,
and we shall then have
y = 2a, 0, 3«, 8a, 18«, etc.
Here x is taken as the independent variable, and y as the
function of x. For each value we assign to x there is a corre-
sponding value of y.
When the relation between the two quantities is expressed
by means of an equation between symbolic expressions, the
one is called an analytic fnuction of the other.
An analytic function is said to be
Explicit when the symbol which represents it stands
alone on one side of the equation;
Implicit when it does not so stand alone.
Example. In the above equation y is an explicit function
of X. But if we have the equation
y''-\-xy = x\
then for each value of x there will be a certain value of y,
which will be found by solving the equation, considering y as
the unknown quantity. Here y is still a function of x, be-
cause to each value of x corresponds a certain value of y; but
because y does not stand alone on one side of the equation it
is called an implicit function.
Remabk. The difference between explicit and implicit
functions is merely one of form, arising from the different
ways in which the relation may be expressed. Thus in the
two forms
VARIABLES AND FUNCTIOm. 5
y — %ax'',
y — 2ax'' = 0;
y is the same function of x; but its form is explicit in the first
and implicit in the second.
An implicit function may be reduced to an explicit one by
solving the equation, regarding the function as the unknown
quantity. But as the solution may be either impracticable
or too complicated for conyenient use, it may be impossible to
express the function otherwise than in an implicit form.
3. Classification of Functions. When y is an explicit
function of x it is, by definition, equal to a symbolic expression
containing the symbol x. Hence we may call either y or the
symbolic expression the function of x, the two being equiva-
lent. Indeed any algebraic expression containing a symbol is,
by definition, a function of the quantity represented by the
symbol, because its value must depend upon that of the sym-
bol.
Every algebraic expression indicates that certain operations
are to be performed upon the quantities represented by the
symbols. These operations are:
1. Addition and subtraction, included algebraically in one
class.
3. Multiplication, including involution,
3. Division.
4. Evolution, or the extraction of roots.
A function which involves only these four operations is
called algebraic.
Functions are classified according to the operations which
must be performed in order to obtain their values from the
values of the independent variables upon which they depend.
A rational function is one in which the only operations
indicated upon or with the independent variable are those of
addition, multiplication, or division.
6 THE DIFFERENTIAL CALCULUS.
An entire function is a rational one in which the only in-
dicated operations are those of addition and multiplication.
Examples. The expression
fls + Ja; + ca;' + dx'
is an entire function of x, as weU as of a, i, c and d.
The expression
, m . c
X x" -\-nx
is a rational function of x, but not an entire function of x.
An irrational function of a Tariable is one in which the
extraction of some root of an expression containing that vari-
able is indicated.
Example. The expressions
V'fls + bx, (a + mx' + «a;')
are irrational functions of x.
Functions which cannot be represented by any finite com-
bination of the algebraic operations above enumerated are
called transcendental.
An exponential function is one in which the variable
enters into an exponent.
Example. The expressions
{a + a;)"", d"
are entire functions of x when n and y are integers. But
they are exponential functions of y.
Other transcendental functions are:
Trigonometric functions, the sine, cosine, etc.
IiOgarithmic functions, which require the finding of a
logarithm.
Circular functions, which are the inverse of the trigo-
nometric functions; for example, if
y = a trigonometric function of x, sin x for instance,
then a; is a circular function of y, namely, the arc of which y
is the sine. . , , •?
VARIABLES AND FUNCTIONS. 7
3. Functional Notation. For brevity and generality we
may represent any function of a variable by a single symbol
having a mark to indicate the variable attached to it, in any
form we may elect. Such a symbol is called a functional
symbol or a symbol of operation.
The most common functional symbols are
F, f and 0;
but any signs or mode of writing whatever may be used.
Then, such expressions as
F{^), A^), 'K^),
each mean
" some symbolic expression containing x."
The variable is enclosed in parentheses in order that the
function may not be mistaken for the product of a quantity
F,foT 0by a;.
Identical Functions. Functions which indicate identical
operations upon two variables are considered as identical.
Example. If we consider the expression
a + ly
as a certain function of y, then
a-\-hx
is that same function of x, and
a + b{x + y)
is that same function olx-\- y.
When the functional notation is applied, then:
Identical functions are represented by the same functional
symbols.
Examples. If we put
F(x) = a + bx,
we shall have F(y) = a + by;
nf) = « + h";
F{x' -f) = a + i{x' - f).
8 THE DIFFERENTIAL OALGULUS.
In general. If we define afunctional symbol as representing
a certain function of a variable, that same symbol applied to
a second variable will represent the expression formed by sub-
stituting the second variable for the first.
In applying tWs rule any expression may be regarded as a
Tariable to be substituted, as, in the last example, we used
x' — y' as a variable to be substituted for x in the original
expression.
EXERCISES.
1. If we put
0(a;) = ax',
it is required to form and reduce the functions
0(y), 0(5), 0(a), <P{-x), 4>{x'), 0(1).
a. Putting
_, , 1 + a;
it is required to form and reduce
^«+')- ^(5). ^(|. ^(1). ^(l)+'(^)-
3. Putting
it is required to form and reduce
/(.-.), /(. + .), /(I), /(£).
4. If 0(a;) = a'x + ex',
form and reduce the expressions
<p{x'), <p{a'), <p{ax), (P{bx), (f>{a + c), fp{a-c).
5. Suppose 0(a;) = ax' — a'x, and thence form
<P{y)> <P{x), (fr{by),
<P{^ + y)> 'P{.x + a), <p{x — a),
(p{x + ay), (f){x - az/), ct>{x').
VARIABLES AND FVNOTIOm. 9
6. Suppose /(a;) = x', and thence form the values of
/(I). A^l, A^\ A^% A^% /(«-•")•
t. Let us put 0(ot) = m{m — l){m — 2) (m — 3); thence
find the values of
0(6), 0(5), 0(4), 0(3), 0(3), 0(1), 0(0), 0(- 1), 0(- 2).
8. Prove that if we put (j){x) = a^, we shall have
0(a; + 2/) = 0(a:) x 0(2^); 0(a;2^) = [0(2;)]" =- [0(2/)]*-
4. Functions of Several Variables, An algebraic expres-
sion containing several quantities may be represented by any
symbol having the letters which represent the quantities at-
tached.
Examples. We may put
0(a;, y) = ax — ly,
the comma being inserted between x and y so that their
product shall not be understood. We shall then have
0(m, n) = am — in,
4>{y, ») = ay — ix,
the letters being simply interchangedj
0(a; -\- y, X - y) = a{x + y) - i{x - y)
= {a- b)x +(a + b)y;
0(a, b) = a' - b'l
0(5, a) = ab — ba = 0;
0(a + b, ab) = a{a -{- b) — ab';
0(«, a) = a' — la;
etc. etc.
If we put 0(a, b, c) = 3a + 35 — 5c, we shall have
0(a;, z, y)=2x + 3z- by;
<l)(z, y, x)=%z-\-^ - 5x;
<j>{m, m, — m) = 2m -\- 3m -\- 5m = 10m;
0(3, 8, 6) = 3-3 + 3-8 - 5-6 = 0.
10 THE nrFFERENTIAL CALCULUS.
EXERCISES.
Let us put <l>{x, y) =3z — 4:y;
f{x, y) = ax + by;
f(x, y, z)=ax + by — abz.
Thence form the expressions:
I- fPiy, ^)- 2- 0K i)' 3' 0(3, 4).
4. 0(4, 3). 5. 0(10, 1). 6. /(a, 5).
7. Ah a). 8. /(y, a;). 9. /(7, - 3).
lO- A<1> —P)- "• /(2, !»» y)- 12- /(*, «J 2).
13- /(«, 5, c)- 14- /(«', 5°, c')- 15- A-a,-i,-ab).
Sometimes there is no need of any functional symbol
except the parentheses. For example, the form (m, n) may
be used to indicate any function of m and n.
EXERCISES.
T J. 4. / \ "»(»* — l){m — 3)
Let us put (to, n) = —) -{-f ^,
^ ' ' n{n — V)(n — 2)
then find the values of —
1. (3, 3). 2. (4, 3). 3. (5, 3).
4- (6, 3). 5. (7, 3). 6. (8, 3).
7. (3, - 1). 8. (3, - 3). 9. (4, - 2).
5. Functions of Functions. By the definitions of the pre-
ceding chapter, the expression
f[<P{^))
will mean the expression obtained by substituting 0(a;) for x
in /(a;).
We may here omit the larger parentheses and write /0(a;)
instead of /( 0(a;) ) .
For example, using the notation of exercises 1 and 3 of
§ 3, we shall have
,,, . ax" — a a;' — 1
ax' + a x' + V
VARIABLES AND FUNCTIONS. 11
For brevity we use the notation
tp^x) = <p{<t>{x)y
Continuing the same system, we have
ct>\x) = cl,[ct>\x)^ =0=(0W);
etc. etc. etc.
EXAMPLI
IS. 1. If
cl){x) = ax\
en
cjy'ix) = a(axy = a'x';
<p\x) = a{a'xy = a'x';
etc. etc. etc.
3. If
f{x) = a-x,
en
f'(x) = a — (a — x) = a
f\x) = a — fix) = a — x;
and, in general,
Eemakk. The functional nomenclature may be simplified
to any extent.
1. The parentheses are quite unnecessary when there is no
danger of mistaking the form for a product.
2. When it is once known what the variables are, we may
write the functional symbol without them. Thus the symbol
may be taken to mean <px or <p{x).
6. Product of the First n Numbers. The symbol n\, called
factorial n, is used to express the product of the first n num-
bers,
1-2-3-...M.
Thus, 1! = 1;
2! = l-2 = 2;
3! = l-2-3 = 6;
4! = l-3-3-4^24;
etc. etc.
12 THE DIFFERENTIAL CALCULUS.
It will be seen that
2! = 2-1!;
3! = 3-21;
and, in general, n\ — n-(n — V)\,
whatever number n may represent.
EXERCISES.
Compute the values of —
I. 5! 2. 6! 3. 8!
7! 8!
^' 3! 4! ^' 3! 5!
6. Prove the equation 2 • 4 • 6 • 8 • . . . 2m = 2"w !
7. Prove that, when n is even,
Mj _ n{n — 2) (w —4)... 4-2
2' ~ »
7 . Binomial Coefficients. The binomial coefiBcient
w(w — 1) (m — 2) to s terms
1-2-3-...S
is expressed in the abbreviated form
(7).
the parentheses being used to distinguish the expression from
the fraction — .
s
EXAMPLES.
3.
7- 6-5-4-3
2-3-4-5
©4=
In\ n
In\ _ n{n — 1) (w — 2)
\3/ "" 1-2-3
VARIABLES AND FVW0TI0N8.
13
EXERCISES.
ProTe the formulse:
I) = (I)-
5\ _ 5!
2/ ~%\ 3]"
'• ^%)=[^^-
n\
n-\-l
n-\-l fn
.5 + 1/ s + 1
n\ . In \ tn -\- 1
s\ {n — s)\
6. (!) + (!)=/« + '
r) + (l) = (-
i)+f
)■
8. Grapliic Representation of Functions. The methods of
Analytic Geometry enable us to represent functions to the eye
by means of curves. The common way of doing this is to
represent the independent variable by the abscissa of a point,
and the corresponding value of the function by its ordinate.
Let a;,, a-,, x^, etc., be
different values of the in-
dependent variable, and
y^> y,> y.> etc., the cor-
responding values of the
function. We lay off
upon the axis of abscis-
sas the lengths OX^,
T
y
(L
Pl-'-T-
Pa
y^
Vi
JCi X2 Xs
ox„ ox,
to X^, X,,
FlQ. 1.
1'
3, etc., equal
X,, etc., and terminating at the points X^, X„ X„
etc. At each of these points we erect a perpendicular to rep-
resent the corresponding value of y. The ends, P,, P^, P^,
of these perpendiculars will generally terminate on a curve
line, the form of which shows the nature of the function.
It must be clearly seen and remembered that it is not the
curve itself which represents the values of the function, but
the ordinates of the curve.
14
THE DIFFERENTIAL CALCULUS.
Fia. 3.
9. Continuity and Discontinuity of Fnrwtions. Let us
consider the graphic repreeentation of a function in the most
geai€Tal"way. "We measure off a series of values, OX,, 0X„
0X3, etc., of the independent variable, and at the points X„
X,, X3, etc., we erect ordinates.
In order that the variable ordinate ^^
may actually be a function of x it is
sufficient if, for every value of the
abscissa, there is a corresponding
value of the ordinate.
N"ow we might conceive of such
a function that there should be no • '' ' 1 ' ' 1 1 ' ' ' 1 ' 1 ' ' '^f-
relation between the different val-
ues of the ordinates, but that every
separate point should have its own
separate ordinate, as shown in
Fig. 2. If this remained true how
numerous soever we made the ordi-
nates, then the ends of the latter would not terminate in any
curve at all, but would be scattered over the plane. Such
a function would be called discontinuous at every point.
Such, however, is not the kind of functions commonly
considered in mathematics. The functions with which we
are now concerned are such that, however irregular they may
appear when the values of x are widely separated, the ends
of the ordinates will terminate in a curve when we bring
those values close enough together.
If a function is such that when the point representing the
independent variable moves continuously from X, to X, (Fig.
1) the end of the ordinate describes an unbroken curve, then
we call the function continuous between the values a;, and
a;, of the independent variable.
If the curve remains unbroken how far soever we suppose
a; to increase, positively or negatively, we call the function
continuous for all values of the independent variable.
VABIABLB8 AND FUNCTIONS.
15
But if there is a value a oix for which there is a break of
any kind in the curve, we call the function discontinuous for
i]ie value a of the independent variaMe.
Let us, for example, consider the funjetioji
6(a — x)'
Let us measure ofl on the axis of abscissas the length OX
= a . Then as we make our varying ordinate approach X
from the left it will increase positively without limit, and the
curve will extend upwards to infinity; if we approach X from
the right-hand side, the ordinate will be negative and the
curve will go downwards to infinity. Thus the curve will not
form a continuous branch from the one side to the other.
Thus the above function is discontinuous for the value a of x.
/
X
Fio. 3.
10. Many-valued Functions. In all that precedes, we
have spoken as if to each value of the independent variable
corresponded only one value of the function. But it may
16
THE DIFFEBMNTIAL CALGULUS.
happen that there are several such values. For example, if y
is an implicit function of x represented by the equation
y' + mxy^ + nx'y -\-px^ — 0,
then we know, by the theory of equations, that there tyill be
three values of y for each value assigned to the variable x.
Def. According as a function admits of one, two or n
values, it is called one-valued, two-valued or w-valued.
Infinitely-valued Functions. It piay happen that to each
value of the variable there are an infinity of different values
of the function. A case of this is the function sin <" '' x, or
the arc of which x is the sine. This arc may be either the
smallest arc which has x for its sine, or this smallest arc in-
creased by any entire number of circumferences.
Take, for example, the arc whose sine shall
be+i
The two smallest arcs will be
30° = \7t and 150° = ^n.
But if we take the function in its most gen-
eral sense it may have any of the values
{%+\)7t; (4 + i);r; (6 + i);r, etc.,
or (2+f);r; (4 + |);r; (6 + |);r, etc.
When we represent an m-valued function
graphically, there will be n values to each ordi-
nate. Hence each ordinate will cut the curve
in n points, real or imaginary.
The figure in the margin represents the infi-
nitely-valued function
« = a sin ^~"— .
When — a <.x < -\-a, any ordinate will cut
ihe curve in an infinity of points.
LIMITS AND INFINITESIMALS. 17
CHAPTER II.
OF LIMITS AND INFINITESIMALS.
1 1 . Limits. The method of limits is an indirect method
of arriving at the values of certain quantities which do not
admit of direct determination. The method rests upon the
following axioms and definition:
Axiom I. Any quantity, however small, may be multiplied
by so great a number as to exceed any other quantity of the
same kind, however great, to which a fixed value is assigned.
Axiom II. Conversely, any quantity, however great, may
be divided into so many parts that each part shall be less than
any other quantity of the same kind, however small, to which
a fixed value is assigned.
Axiom III. Any quantity may be divided into any num-
ber of parts ; or multiplied any number of times.
Def. The limit of a variable quantity X is a quantity L,
which we conceive Xto approach in such a way that the dif-
ference L — X becomes less than any quantity we can name,
but which we do not conceive X to reach.
Example. If we have a variable quantity X and a con-
stant quantity L, and if X, in varying according to any mathe-
matical laWj takes the successive values
L ± 0.1,
L ± 0.01,
L ± 0.001,
L ± 0.0001,
and so on indefinitely, without becoming equal to L, then we
say that L is the limit of x.
18 THE DIFFERENTIAL CALCULUS.
13. Infinites and Infinitesimals. Definitions.
1. An infinite quantity is one considered as becoming
greater than any quantity which we can name.
2. An infinitesimal quantity is one considered in the
act of becoming less than any quantity which we can name;
that is, in the act of approaching zero as a limit.
3. A finite quantity is one which is neither infinite nor in-
finitesimal.*
Examples. If of a quantity x we either suppose or prove
X > 10,
X > 100,
X > 100000,
and so on without end, then x is called an infinite quantity.
If of a quantity h we either suppose or prove
h < 0.1,
h < 0.001,
h < 0.00001,
and so on without end, then h is an infinitesimal quantity.
The preceding conceptions of limits, infinites and infinitesi-
mals are applied in the following ways: Let us have an inde-
pendent variable x, and a function of that variable which we
call y.
Now, in order to apply the method of limits, we may make
three suppositions respecting the value of x, namely:
1. That X approaches some finite limit.
2. That X increases without limit (i.e., is infinite).
3. That X diminishes without limit (i.e., is infinitesimal).
In each of these cases the result may be that y approaches
a finite limit, or is infinite, or is infinitesimal.
* Strictly speaking, the words infinite and infinitesimal are both adjec-
tives qualifying a qiuintity. But the second has lately heen used also aa
a noun, and we shall therefore use the word infinite as a noun meaning
infinite quantity.
LIMITS AND INFINITESIMALS. 19
For example, let us have
x-\-a
y = — ■ — •
" X — a
Then—
When X approaches the limit a, y becomes infinite.
'When X becomes infinite, y approaches the limit + !•
When X becomes infinitesimal, y approaches the limit — 1.
The symbol i, followed by that of zero or a finite quantity,
means ''approaches the limit." The symbols ioo mean
"increases without limit" or "becomes infinite." Hence
the three last statements may be expressed symbolically, as
follows:
x-\- a ,
When X i a, then
X — a
When a; = 00, then == + 1:
X — a
etc. etc.
The same statements are more commonly expressed thus:
,. x-\- a . . .
lim. — ■ — (x = a) = 00 :
X — a^ '
lim. — ■ — (a; = 00 ) = +1:
X — a^ '
lim. 5+L? (^ ^ 0) = - 1.
X — a^ '
13. Properties of Infinite and Infinitesimal Quantities.
Theoebm I. The product of an infinitesimal by any finite
factor, however great, is an infinitesimal.
Proof. Let /* be the infinitesimal, and n the finite factor
by which it is multiplied. I say how great soever n may be,
7ih is also an infinitesimal. For, if tih does not become less
than any quantity we can name, let a be a quantity less than
which it does not become. Then if we take, as we may,
h < ~, (Axiom III.)
n ^ '
we shall have nh < a.
20 THE DIFFERENTIAL CALCULUS.
That is, nh is less than a and not less than a, which is
absurd.
Hence oih becomes less than any quantity we can name,
and is therefore infinitesimal, by definition.
Theoeem II. The quotient of an injitiite quantity hy any
finite divisor, hotoever great, is infinite.
Proof. Let X be the infinite quantity, and n the finite
divisor. It X -i- n does not increase beyond every limit, let
K be some quantity which it cannot exceed. Then by taking
• X>n£:, (Ax. III.)
we shall have — > X;
n
that is, — greater than the quantity which it cannot exceed,
which is absurd.
Hence X -^ n increases beyond every limit we can name
when X does, and is therefore infinite when X is infinite.
Theoeem III. The product of any finite quantity, how-
ever small, by an infinite multiplier, is infinite.
This foUows at once from Axiom I., since by increasing the
multiplier we may make the product greater than any quan-
tity we can name.
Theoeem IV. The quotient of any finite quantity, how-
ever great, iy an infinite divisor is infinitesimal.
This follows at once from Axiom II., since by increasing
the divisor the quotient may be made less than any finite
quantity.
Theoeem V. The reciprocal of an infinitesimal is an in-
finite, and vice versa.
Let h be an infinitesimal. If :r is not infinite, there must
A
be some quantity which we can name which t- does not ex-
LIMITS AND INFINITESIMALB. 21
ceed. Let K\i& that quantity. Because h is infinitesimal,
we may have
which gives j-> K;
that is, 7- greater than a quantity it can never exceed, which
is absurd.
The converse theorem may be proved in the same way.
14. Orders of Infinitesimals. Def. If the ratio of one
infinitesimal to another approaches a finite limit, they are
called infinitesimals of the same order.
If the ratio is itself infinitesimal, the lesser infinitesimal is
said to be of higher order than the other.
Theorem VI. If we have a series proceeding according
to the powers of h,
A+Bh+Ch' + Dh' + etc.,
in which the coefficients A, B, G, are all finite, then, if h be-
comes infinitesimal, each term after the first is an infinitesi-
mal of higher order than the term preceding.
Proof. The ratio of two consecutive terms, the third and
fourth for example, is
Dh' _D
G¥~ G '
D
By hypothesis, Cand D are both finite; hence ^ is finite;
hence when h approaches the limit zero, ^/j becomes an in^
finitesimal (§13, Th. I.). Thus, by definition, the term Dh'
is an infinitesimal of higher order than G¥.
Def. The orders of infinitesimals are numbered by taking
some one infinitesimal as a base and calling it an infinitesi-
mal of the first order. Then, an infinitesimal whose ratio to
22 THM DIFFERENTIAL CALCULUS.
the wth power of the base approaches a finite limit is called
an infiiiitesimal of the nth order.
Example. If A be taken as the base, the term
Bh is of the first order ' .■ Bh:h — the finite quantity B;
CW " " second" • .■ Ch' : h' = " " C;
Eli^ " " ?tth " • . • Eh"" : A" = " " E.
Cor. 1. Since when « = we have BJi^ = Bh" = B for
all values of h, it follows that an infinitesimal of the order
zero is the same as a finite quantity.
Cor. 2. It may be shown in the same way that the product
of any two infinitesimals of the first order is an infinitesimal
of the second order.
15. Orders of Infinites. If the ratio of two infinite
quantities approaches a finite limit, they are called infinites
of the same order.
If the ratio increases without limit, the greater term of the
ratio is called an infinite of higher order than the other.
Theorem VIL In a series of terms arranged according
to the powers of x,
A+Bx+Ox' + Dx' + etc.,
if A, B, C, etc., are all finite, then, when x becomes infinite,
each term after the first is an infinite of higher order than the
term preceding.
For, the ratio of two consecutive terms is of the form -^.x,
x>
which becomes infinite with x (Th. III.).
Def. Orders of infinity are numbered by taking some one
infinite as a base, and calling it an infinite of the first order.
Then, an infinite whose ratio to the nth. power of the base
approaches a finite limit is called an infinite of the «th order.
Thus, taking x as the standard, when it becomes infinite
we call Bx infinite of the first order, Gx' qI the second order,
etc.
LIMITS AND INFINITESIMALS. 23
NOTE ON THE PRECEDING CHAPTERS.
In beginning the Calculus, conceptions are presented to the student
which seem beyond his grasp, and methods which seem to lack rigor.
Really, however, the fundamental principle of these methods is as old
as Euclid, and is met with in all works on elementary geometry which
treat of the area of the circle. The simplest form in which the princi-
ple appears is seen in the following case.
Let us have to compare two quantities A and B, in order to determine
whether they are equal. If they are not equal, then they must differ by
some quantity. If, now, taking any arbitrary quantity Ti, we can prove
that
A-B<h
wUJumt making a/ny supposition respecting the valiis of h, this will show
that A and B are rigorously equal. For if they differed by the quantity
a, then when /i was less than a the above inequality would not hold
true. But as we have been supposed to prove it for all vnlues of h, it
must be true when h is less than a. In this case h might be considered
an infinitesimal, although in the Elements of Euclid it is represented on
the page of the book by a figure nearly an inch square.
Infinitesimal quantities were formerly called infinitely small. When
they were introduced by Leibnitz many able mathematicians were unable
to accept them. Bishop Berkeley wrote several essays against them, in
one of which he suggested that they might be called the ghosts of departed
quantities. The following propositions are presented in the hope that
they may save the student unnecessary efforts of thought in the study of
this subject.
Firstly, there is no need that a quantity should be considered as ab-
solutely infinite. A mathematical magnitude, considered as a quantity,
must in its very nature have boundaries, because mathematics is con-
cerned with the relation between magnitudes as greater or less, and
we can compare two magnitudes as greater or less only by comparing
their boundaries. An absolutely infinite magnitude, having no boun-
daries to compare, cannot be compared with anything.
Secondly, it is equally unnecessary to suppose the existence, either in
nature or in thought, of quantities which are absolutely smaller than
any finite quantity whatever.
24 THE DIFFERBNTIAL CALCULUS.
But however small a quantity may be, there may always be another
still smaller in any ratio. Hence, although it is perfectly true that no
quantity can be otherwise than finite, yet it is equally true that a quantity
may be less or greater than any fixed quantity we may name.
Both infinite and infinitesimal quantities are therefore essentially in-
definite, because by considering them in the act of increasing beyond, or
decreasing below, every assignable value, we do away with the very pos-
sibility of assigning values to them. They are used only as auxiliaries
to lead us to a knowledge of finite quantities, and their magnitudes are
never themselves the object of investigation.
The essentially indefinite nature of infinites and infinitesimals may be
illustrated as follows:
If we have an equation of the form
ct.
then for every pair of finite values we assign to a and b there will be a
definite value of x.
But if we suppose A and S to be infinite, and at the same time inde-
pendent of each other, there will be no definite value to x. Considering
both terms as absolutely infinite, they will have no bounds, and there-
fore cannot be compared in value. Considered as increasing without
limit, one may be any number of times greater than the other, and thus
the fraction may have any value we choose to assign it. Seeking for
the value of such a fraction is like trying to answer the old question
concerning the effect of an irresistible force acting upon an immovable
obstacle.
DIFFEBENTIALa AND DEBIVATI7M8. 25
CHAPTER III.
OF DIFFERENTIALS AND DERIVATIVES.
16. Def. An increment of a variable is the difference
between two values of that variable.
An equivalent definition is: An increment is a quantity
added to one value of a variable in order to obtain another
value.
Notation. An increment is expressed by the symbol A
written before the symbol of the variable.
Example. If we have the different variables
X, y, u,
and the increments Ax, Ay, Au,
other values of the variables will be
X + ^x, y + Ay, u + Au.
Here A is not a factor multiplying x, but a symbol meaning
"increment of," or, in familiar language, "a little piece of."
In considering the respective increments of an independent
variable, and of its function, the following five quantities
come into play and are each to be clearly conceived.
1. A value of the independent variable, which we may take
at pleasure.
2. The corresponding value of the function, which will be
fixed by that of the independent variable.
3. An increment of the independent variable, also taken at
pleasure.
4. The corresponding increment of the function, deter-
mined by that of the independent variable.
5. The ratio of these increments.
26
THE DIFFERENTIAL CALCULUS.
To represent these quantities, let the relation between the
variable x and the function y be expressed by a curve. Let
OX be one value of x, and OX' another. Let XP and X'P'
be the corresponding values of y, leading to the points P and
P' of the curve. We shall then have —
1. OX = X, a value of the independent variable.
2. XP = y, the corresponding value of the function.
3. XX = Ar, an arbitrary increment of x.
4. RP' = Ay, the corresponding increment of y.
All
5. Then, by Plane Trigonometry, the quotient -~-_ will be
the tangent of the angle PQX; that is, the tangent of the
angle which the secant PP' makes with the axis of abscissas.
Thus we have geometrical representations of the five fun-
damental quantities under consideration.
17. First Idea of Differentials ajid Derivatives. Let us
take, for illustration, the function
y = nx'. (1)
Giving to x the increment Ax, the new value of y will be
71 {x + Axf.
Hence y -\- Ay = n{x + Ax)' = nx' -\- %nxAx -\- nAx', (2)
DIFFBBBNTIAL8 AND DEBIVATIVES. 27
Subtracting (1) from (2), we have, for the increment of y,
Ay = n{2x + Ax)Ax, (3)
Because, when Ax becomes infinitesimal,
lim, (3a; + Ax) = 2x,
we hare, for the ratio of the increments,
-£ = 2,ix + nAx, (4)
and, when Ax becomes infinitesimal,
lim. ^ = 'Hnx. (5)
Def. The differential of a quantity is its infinitesimal
increment; that is, its increment considered in the act of ap-
proaching zero as its limit, or of becoming smaller than any
quantity we can name.
Notation. The differential of a quantity is indicated by
the symbol d written before the symbol of the quantity.
For example, the expressions
dx, du, d(x + y),
mean any infinitesimal increments of x, u, (x -\- y), respect-
ively.
Thus the substitution of d for A in the notation of incre-
ments indicates that the increment represented by A is sup-
posed to be infinitesimal, and that we are to consider the limit
toward which some quantity arising from the increment then
approaches.
Using this notation, the equation (5) may be written
-^-- — %nx.
dx
We also express this value of the limiting ratio in the form
dy = 2nxdx;
meaning thereby that the ratio of the two members of this
equation has unity as its limit. This is evident from Eq. (3).
28
THE DIFFERENTIAL CALCULUS.
Def.
If « is a function of x, the ratio -~- of the differential
" dx
of y to that of x is called the derivative of the function, or
the derived function.
18. Illustrations. As the logic of infinitesimals ofEera
great difficulties to the beginner, some illustrations of the
subject may be of value to him.
Consider the following proposition:
The error introduced hy neglecting all the powers of an in-
crement above the first may be made as small as we please by
diminishing the increment.
Let us suppose n = %m the equation (1). We then have
the equations
y = 2a;';
Ay^i-cAx-\-%Ax'; _ ^^^
^ = 4a; + 'HAx.
The ratio of the two terms of the second member is
%Ax Ax
-^' '' W
Let us now neglect this quantity and write the erroneous
equation
Ax
If, now,
we
suppose
Ax <
Ax <
Ax <
X
Too'
X
ioooo'
X
1000000'
etc..
the equation
(b) will still be
true within
J_
300
1
•^0000
1
2000000
etc.
part;
part;
part;
So long as we assign any definite value to Ax, it is clear
that there will be some error in neglecting Ax. But there is
no error in the equations
dy = ixdx and — = 4a;,
ctss
DIFFERENTIALS AND DERIVATIVEB. 29
provided that we interpret them as expressing the limit which
Ay
-p approaches as Ax approaches the limit zero, and interpret
all our results accordingly.
19. Illustration by Velocities. Let us consider what is
meant by the familiar idea of a train which may be contin-
ually changing its speed passing a certain point with a certain
speed. To fix the ideas, suppose the train has just started
and is every moment accelerating its speed in such manner
that the total number of feet it has advanced is equal to the
square of the number of seconds since it started. Put
S = the distance travelled expressed in feet;
t = the time expressed in seconds.
We shall then have S = f,
and for the distances travelled:
Number of seconds, 0; 1; 2; 3; 4; 6; etc.;
Distance travelled, 0; 1; 4; 9; 16; 35; etc.;
Distance in each second, 1; 3; 5; 7; 9; 11; etc.
B
ill 8 I 5 I T I 9 1 11 I
Fio. 6.
Let this line represent the space travelled the first five
seconds from the starting time, and let us inquire with what
velocity the train passed the point B at the end of 4°.
Since distance travelled = velocity x time, the mean ve-
locity is found by dividing the space by the time required to
pass over that space. Now, the train had travelled
16 feet in the time 4 seconds,
and (4 + Aty feet in (4 + At) seconds,
or 16 + 9,At + Af feet in (4 + At) seconds.
Subtracting 16 feet and 4 seconds, we see that in the time
At after the end of the 4 seconds the train went %At -\- At*
= As feet. Hence its mean velocity from 4° to 4' + At is
30
THE DIFFERENTIAL OALCULUS.
As
At
= (8 + At) feet per second.
Now it is clear that, since the train was continually accel-
erated how small soever we take At, the mean velocity during
this interval will exceed that with which it passed B. But
it is also clear that by supposing At to approach the limit zero,
we shall approach the required velocity as our limit. Hence
the velocity with which B was passed is rigorously
ds
di
— 8 feet per second.
Fig. 7.
30. Geometrical Illustration. If, in the figure, we sup-
pose the point P' to approach P as its limit, the increments
Ax and Ay will approach the limit zero, and the secant P'P
will approach the tangent at the point P as its limit. We
have already shown that
Ax
tangent of angle made by secant with axis of abscissas.
Passing to the limit, we have the rigorous proposition
-^ = tangent of angle which the tangent at the point P
makes with the axis of abscissas.
dx
DIFFERBNTTATION OF EXPLICIT FUNOTIONB. 31
CHAPTER IV.
DIFFERENTIATION OF EXPLICIT FUNCTIONS.
31. Def. The process of finding the differential and the
derivative of a function is called differentiation.
As exemplified in §§ 16, 17, it may be generalized as fol-
lows: We have given
(1) An independent variable = x.
(2) A function of that variable = <t>{x).
(3) We assign to x an increment = Ax; whereby ^{x) is
changed into (p{x -\- Ax).
(4) We thus have 0(a; -(- Ax) — (t>{x) as the increment of
4>{x). We may put
A(j}{x) B 4){x + Ax) — <p(x).
(5) We then form the ratio
A(p{x)
Ax
(«)
and seek its limit when Ax becomes infinitesimal. Using the
notation of the last chapter, we have
flcf){x) ,. A<p{x) , . . „,
which is the derivative of 0(a;).
In order to find the ratio {a), it is necessary to develop
cl){x -\- Ax) in powers of Ax to at least the first power of Ax.
Let this development be
0(a; -f Ax) = X„ + X^Ax + X^Ax' + (1)
In the second member of this equation X^, X„ etc., will be
functions of x; and it is evident that X„ can be nothing but
32 TEB DIFFERENTIAL CALOXTLUS.
(p{x) itself, because it is the value of <p{x + //») when Jx — 0.
Thus we have
J0(a;) = <p{x + Ax) - cl>(x) - (X, + XJx) Jx -\- . . . ;
Passing to the limit,
d<p{x) = X/lz;
M£) = x, (3)
Thus, by comparing with (1), we have the following:
Theokem I. The derivative of a function is the coefficient
of the first power of the increment of the independent variable
when the function is developed in powers of that increment.
If we have to differentiate a function of several variable
quantities, x, y, z, etc., we assign an inprement to each vari-
able, and develop the function in powers and products of the
increments.
Subtracting the original function, the remainder will be its
increment.
The terms of highest order in this increment, considered
as infinitesimals, are then called the differential of the
function.
The following are the special cases by combining which all
derivatives of rational functions may be found.
33. Differentials of Sums. Let x, y, z, u, etc., be any
variables or functions whatever. Their sum will be
x-\-y-\-z-\-u-\- etc.
Assigning to each an increment, x will become x + Ax, y
will become y -\- Ay, etc. Hence the sum will become
X -\- Ax -\- y -\- Ay -\- z -\- Az -\- u -\~ Au -\- etc.
Subtracting the original expression, we find the increment of
the sum to be
Ax + Ay -\- Az -\- Au -\- etc.
DIFFEBENTIATIOW OF EXPLICIT FUNCTIONS. 33
Hence, when the increments become infinitesimal,
d{x + y + ^ + w + etc.) = dx -{- dy -\- dz -\- du -\- etc., (3)
or, in words:
Theorem II. The differential of the sum of any number
of variables is equal to the sum of their differentials.
In this theorem the quantities x, y, z, u, etc., maybe either
independent variables, or functions of one or more variables.
83. Differential of a Multiple. Let it be required to find
the differential of
ax,
a being a constant.
Giving X the increment Ax, the expression will become
a{x -\- Ax).
Then, proceeding as before, we find
d{ax) = adx. (4)
34. Theorem III. The differential of any constatit is
zero.
For, by definition, a constant is a quantity which we sup-
pose invariable, and to which we cannot, therefore, assign any
increment.
We therefore have, from Theorem I. when a; is a variable
and a is a constant,
d{x -|- a) = (?a; -)- = dx,
or, in words:
Theorem IV. The differential of the sum of a constant
and a variable is equal to the differential of the variable alone.
Eemark. It will be readily seen that the conclusions of
§§ 23-24 are equally true whether we suppose the increments
to be finite or infinitesimal. This is no longer the case when
powers or products of some finite increments enter into the
expression for other finite increments.
34 TSE DIFFERENTIAL CALCULUS.
EXERCISBS.
It is required, by combining the preceding processes, to
form the difEerentials of the following expressions, supposing
a, b and c to be constants, and all the other literal symbols
to be variables.
I.
U — V.
2. 2u — V.
3-
5-
7-
9-
V -\-x-\-c.
a'x + b-y + c.
iax -J- 5bx — y.
Sx — a -\- ab.
4. ax -\- by.
6. 3a; + 4a«/ + h.
8. &bx — abc.
10. abx — abt.
11.
13-
c{2x + a).
ac{bu + «2;).
12. a{bx + ac).
14. 5c(2aa; — 3by).
IS-
x-y-z.
16. — aa; — by — cz.
17-
19.
— a{bx — cy).
X
a'
18. - b{%ax - Zcv).
20. =^ + f-^
21.
(a + b + c){s
+ t + 3u-
-%)•
22.
{a + 2b' + 3c'
. fau
^ \ c
bv
a
+ 2 si"
35. Differentials of Products and Powers. Take first
the product of two variables, which we shall call u and v.
Then
Product = nv.
Assigning the increments Ait, and Av, the product becomes
(m + All) {v + Ji') = uv + «'^w + M-^« + AuAv.
Subtracting the original function, nv, we find
A{uv) — vAu + (?« + An) Av.
Supposing the increments to become infinitesimals, the co-
efficient of Av in the second member will approach u as its
limit. Hence, passing to the limit (§ 14),
d{uv) = vdu + udv.
JDIFI'BBENTIATION OF EXPLICIT FUNCTIONS. 36
To extend the result to any number of factors, let P be the
product of all the factors but one, and let the remaining fac-
tor be X, so that we have
Product = Px.
By what precedes, we have
d{Px) = xdP + Pdx.
Supposing P to be a product of the two variables u and v,
this result gives
d{uvx) = xd{vii) + uvdx = vxdu -\- uxdv -\- uvdx. («)
If we add a fourth factor, y, we shall have
d{uvxy) — yd{uvx) -\- uvxdy.
If we substitute for d{uvx) its value (a), we see that we
pass frorn. the one case to the other by (1) multiplying all the
terms of the first case by the common factor y; (3) adding
the product of dy into all the other factors.
We are thus led to the conclusion:
Theoeem v. The differential of the product of any num-
ler of variables is equal to the sum of the products formed by
replacing each variable by its differential.
Corollary. If the n factors are all equal, their product will
become the wth power of the variable, and the n difEerentials
will all become equal. Hence, when n is an integer, we have
the general formula
d^x^) = 2-"~VZa; + x'^~\lx -\- etc., to ?i terms,
or dipf') = iix'^~'^dx.
By combining the preceding processes we may form the
differential of any entire function of any number of variables.
EXAHPLES.
I. d{ax + ixy -\- cxyz)
= d{ax) + d{bxy) + d{cxyz) (Th. II. 32)
= adx + bd(xy) + cd{xyz) (Th. III. 23)
= adx -\- b{ydx -\- xdy) + c{yzdx -{- xzdy -j- xydz)
= (a -\- by -\- cyz)dx + {bx + czz)dy + cxydz.
36 THE DIFFERENTIAL GALGULUS.
2. d{ax' + b) = d{ax') (Th. IV.)
= ad{x') (§ 23)
= Sax'dx. (Th. V., Cor.)
3. d{axY) = ad{xY) (§23)
= a[i/"d{x') + «'%")] (Th. V.)
= 3a«/ V Ja; + nax'y " - ^dy. (Th. V. , Cor. )
4. d{a+ x')"- = n{a + a;=)"-'<?(a + x'')=2n{a + x')'—^xdx.
EXERCISES.
Form the differentials of the following expressions, suppos-
ing the letters of the alphabet from a to ii to represent con-
stants:
1. a-\-bx^ -{- ex*. Ans. {2bx -\-4tcx')dx.
2. B+Cy-{- Dy\ 3. axy.
4. bxyz. 5. a(x -\- yz).
6. a{x' + buv). 7. axy -\- buv.
8. h{x'y + xy'). 9. ax'y'.
10. Jx'y'*. II. aSa;'*/'' + Aw^z;"-
12. 2(«ia: + my). 13. (r + ^) (^ + ^r).
14. n{a — a;'). 15. aa;' — byz.
16. (« + 3;) (3 - 2/). 17. (a + x'){b-y').
18. (a — a;) (« — a;'). 19. a;(a + a;) (5 — a;').
20. (^ + 5a; + fe") (2/ + z). xy
21. (^+52/'+Cy)(fl2^+te) "• t-
23- (« + 5w*) (cx^ — ny''). X — uv
25. (a — x){b — a;')(c — x'). ' a
. x-uv. , , 27. x{x' + y{a-x)\.
26. (M + W).
a ^ '
29. {ay' ~ bx') {x - y).
^ U bJ\a '^ b)' 32. «(«+a;)'.
33. (ff + a:?/)'. 34. (aa; + 5y)'.
DIFFERENTIATION OF EXPLICIT FUNCTIONS. 37
36 Differential of a Quotient of Two Variables. Let the
variables be x and y, and let q be their quotient. Then
X
and qy = x.
Differentiating, we have
ydq + qdy = dx.
Solving so as to find the value of dq,
_dx — qdy _ ydx — xdy
^ ~ y ~ ? '
Hence:
Thbokbm VI. The differential of a fraction is equal to
the denominator into the differential of the numerator, minus
the numerator into the differential of the denominator, divided
ly the square of the denominator.
Eemakk. If the numerator is a constant, its difEerential
vanishes, and we have the general formula
d— — ,dx.
X X
EXERCISES.
Form the differentials of the following expressions:
X a -\- X
2.
a
+ y
a
— X
a
-y
a
1*
a
-\-l)X
a
+ iy
X
+ y
« + :
x'
{b + yf
m -\- nx^
8. ;.
m — nx
lO. — 5-i — %
X — y mx — ny^
38 THE BIPFEBENTIAL CALCULUS.
a -\- bx + ex'
VI + xy
'^' m - xY
a , b
a
17
19.
xy + x'y
^' + ?/'
12.
x + yz
y + xz
14.
11
X X
m n
16.
x'~y'
1 1
18.
X y
a
x'-y'
-f x'+f
37. Differentials of Irrational Expression!!. Let it be re-
quired to find the difEerential of the function
m
U = X",
m and n being positive integers. Eaising both members of
the equation to the wth power, we have
«" = a;™.
Taking the differentials of both members,
nu"~^du = mx^^~^dx,
whence
du_mx'"~^_m re"'"' _vi a;*""' _m—-i
dx~n v^"- ~ n /"^-"i ~ n "■" -^ ~ « ^ " ' ^"'
yx"! X "
a formula which corresponds to the corollary of Theorem V.,
where the exponent is entire.
Next, let the fractional exponent be negative. Then
_!1 1
U = X » = -— -,
and, by Th. VI.,
d\x^l mx" dx in -JS-i ,
f^« = ^ =~ n ■ ^"~ "^ ~ w"* " '^'^■'
X " a; »
and, for the derivative,
du _ m ---1
dx~~ n
DIFFERENTIATION OF EXPLICIT FUNCTIONS. 39
From this equation and from {a) we conclude:
Thboebm VII. The formula
d{x'') = nx^'-^dx
liolds true whether the exponent n is entire or fractional, posi-
tive or negative.
We thus derive the following rule for forming the differen-
tials of irrational expressions:
Express the indicated roots by fractional exponents, positive
or negative, and then form the differential by the preceding
methods.
Examples.
1. dVa-\-x = d(a -A- a;)* = Ma -1- a;) - idx = —. — ; — r-.
^ ' ' ^\ 1 / 2(a-{-x)i
2. d^-^, = d[b(a + x)-i] =bd(a + x)-i
3. d(a + bx')i = i(a 4- bx') - * 2bxdx = -. — , , ,^ -.dx,
^ ' " ' (a-{-bx)*
EXERCISES.
Form the differentials of the following expressions:
I.
Va+x.
Va - x\
a
2.
5-
8.
II.
14.
Vb-x.
3-
6.
9-
12.
15-
Va - hx.
4-
Va - bx\
b
Vx + y.
b
lO.
13-
Vx + y'
{a + x)i.
X Va-\-x.
Va + bx' •
{x-a)h
xVa — x.
Va - W
(bx' - a)h
y' Va - by'.
Find the values of -r- in the following cases:
1 6. z« = mx -\ — . 17, M = {mx' — w)*.
40 THB DIFFERENTIAL CALCULUS.
1 8. «t = Vax -\- bx'.
a
+ ca;'
20. u = xVa— X.
a 4- X
21. u = x' Vx' + a.
a — X
a — X a-\- X
38. Logarithmic Functions. It is required to differentiate
the function
u = log X.
We have
Jti, - log {x + Ax) - log a; = log ^-—^- = log (l + "^j-
It is shown in Algebra that we Lave
log (1 + h) = M{h - W + W - etc.),
M being the modulus of the system of logarithms employed.
Ax
Hence, puting — ^ for h, we find
A a
x\
1 Ax
% X
+
3 x'
1
■ etc.
and.
passing to the limit.
du =
Mdx
' X '
du
d'x ''
_M
x'
In the Naperian system ilf=: 1. In algebraic analysis,
logarithms are always understood to be Naperian logarithms
unless some other system is indicated. Hence we write
d-\os a; 1 - - dx
-^— =: — ; d-iogx = — .
dx X ° X
Example,
, - d(axy) axdy + ai/dx dy , dx
d-\os axy = --^^ — — = ^— ^ — - — = — H .
axy axy y x
Remabk. We may often change the form of logarithmic
BIVVEUENTIATION OF EXPLICIT FUNCTIONS. 41
functions, so as to obtain their differentials in various ways.
Thus, in the last example, we have
log {axy) = log a + log x + log y,
from which we obtain the same differential found above. The
student should find the following differentials in two ways
when practicable.
EXERCISES.
Differentiate:
I. log (a -\- x). Ans. . 2. log {x — a),
3. log(:i;' + J'). 4. log (x= - S).
5. log mx. 6. log mx^.
7, log (aa;" + V). 8. log m".
9. log (x + y). 10. log {x - y).
II. log a;y. 12. log (x' + ^z")-
13. log (« + &)". 14. log^.
^ x4- a ^ , a — x
115. log r-^. ID. log 7 .
1-1. y log X. 18. log (« — ar)".
39. Exponential Functions. It is required to differentiate
the function
u = a",
a being a constant.
Taking the logarithms of both members,
log u = X log a.
Differentiating, we have, by the last article,
c?-log u = — = dx log a.
42 THE DIFFERENTIAL CALCULUS.
Hence du — u log a dx^ a" log a dx;
^^ = a^loga,
which is the required derivative.
If a is the Naperian base, whose value is
e = 2.71838 . . . . ,
we have log a = 1. Hence
d-e" _ ^
dx ~
Hence the derivative of e' possesses the remarkable prop-
erty of being identical with the function itself.
EXERCISES.
Differentiate:
I. a^. Ans.
2a===
log
a dx.
2,
a"^.
3- c»+'".
4-
ga + n._
5-
Jl"^ + 'iV^
6. h-^'-y.
7-
/t-*^.
8.
d'ay.
9. a'bK
10.
a'^b'K
II.
ak^b-"".
12. e* + ».
13-
e'e^".
14.
gOat + by^
30. The Trigonometric Functions.
The Sine. Putting h for the increment of x, we have, by
Trigonometry,
sin {x -\- h) — sin a; = 3 cos {x -\- ^h) sin ^h.
Now, let h approach zero as its limit. Then,
sin {x-\-h) — sin x becomes d sin x;
h becomes dx, because it is the increment of x;
cos {x -\- \h) approaches the limit cos x;
sin \h approaches the limit ^U or ^dx, because when
an angle approaches zero as its limit, its ratio to its sine
approaches unity as its limit (Trigonometry).
Hence, passing to the limit,
d'sin X = cos xdx.
DIFFERENTIATION OF EXPLICIT FUNCTIONS. 43
The Cosine. By Trigonometry,
cos {x -\- It) — cos X = — sin {x -\- ^h) sin ih.
«
Hence, as in the case of the sine,
d cos X = — sin x dz.
Taking the derivatives, we have
d sin x
dx
d'cos X
dx
— sin X.
M N
Fig. 8.
PB = A sin r.
KP — A cos X.
Oeometrical Illustration. In
the figure, let OX bo the unit- o
radius. Then, measuring lengths
in terms of this radius, wo shall have
NK = sin x; MB = sin {x + li) ;
ON = cos x; OM = cos (x + h);
Also, supposing a straight lino from K io B,
PK= - KP = KB sin PBK;
PB = KB cos PBK.
When B approaches K as its limit, the angle PBK ap-
proaches XOK, or x, as its limit, and the line KB becomes
dx. Hence, approaching the limit, we find the same equa-
tions as before for d sin x and d cos x.
It is evident that so long as the sine is positive, cos x di-
minishes as X increases, whence <f 'cos x must have the nega-
tive sign.
The Tangent. Expressing the tangent in terms of the sine
and cosine, we have
sin X
tan X = .
Differentiating this fractional expression,
cos xd'sin x — sin xd'cos x sin" xdx + cos' xdx
d tan X = 5 ' — ^
cos X
= sec" xdx,
which is the required differential.
COS X
44 THE DIFFERENTIAL CALCULUS.
We find, by a similar process,
, , - cos a; , , dx
a cot X = d'—. — = — CSC xdx = -r-^- :
sm X Bin X
1 ff'cos a; sin xdx
rf'sec X = d
cos a; cos a; cos a;
= tan X sec a;rfa;;
(?"cosec X — — cot a; esc xdx.
EXERCISES.
Differentiate:
I. cos {a + 2/). z. sin (5 — y). 3. tan (c + 2).
4. sin y cos 2. 5. tan u cos v. 6. sin m tan v.
7. sin aa:. 8. cos ay. 9. tan mz.
10. sin (7* + my), ri. cos (A + my). 12. sin (A — my).
13. cos' a; • [d'cos' x = 2 cos a;<?-cos a; = — sin 2xdx].
14. sin" a;. 15. sin' y. 16. sin' m2.
sin X „ sin' a; cos' x
17. . 18. -. 19. -^-j—.
cos y COB y sm y
20. Show that (?(sin' y + cos' y) = 0, and show why this
result ought to come out by § 34.
21. Differentiate the two members of the identities
cos {a-\- y) — cos a cos y — sin a sin y,
sin {a -\- z) = cos a sin 2 -|- sin a cos 2,
and show that the differentials of the two members of each
equation are identical.
22. Show that d'log sin x = cot x dx;
d'log cos a; = — tan x dx.
31. Circular Functions. A circular function is the in-
verse of a trigonometric function, the independent variable
being the sine, cosine, or other trigonometric function, and
the function the angle. The notation is as follows:
If y ^= sin 2, we write 2 = sin '~ " y or arc-sin y;
If M = tan X, we write x = tan '~ " u or arc-tan u;
etc. etc. etc.
DIFFBRENTIATION OF EXPLICIT FUNCTIONS. 45
Differentiation of Circular Functions. If we have to dif-
ferentiate
z = sin '~ " y,
we shall have
y = sin z; dy ~ cos z dz = Vl — sin" a dz;
• dz = ^^^ = = ^^ («)
Vl - sin'' 2 +^1 - .v"'
TAe Inverse Cosine. If « be the inverse cosine of y, we
find, in the same way,
dz=- -t=. ih)
The Inverse Tangent. If we have
z = tan '■" " y;
then, y = tan z; dy = sec" zdz = (1 -{- tan" «)<?«;
The Inverse Cotangent. We find, in a similar way,
^ 1 + 2/
y^e Inverse Secant. If we have
z = sec<~'' y;
{d}
then, y = sec z; dy = tan z sec zdz = y Vy' — 1 <Zz;
...dz^-^^=. (e)
yVf-1
The Inverse Cosecant. We find, in a similar way,
«?-csc<-"V= ,
yVy'-l
46 TSM DIFFERENTIAL CALCULUS.
EXERCISES.
Differentiate with respect to x or z:
2. cos<~" (« + a).
3-
sin*-" (mx + a).
4-
COS*-" -.
X
S-
tan <-»(«--).
\ zJ
6.
tan<-»f2+-\
7-
''"'-L-+I)-
8.
tan<-" («').
9-
sec '~ '^ 2-1 — .
lO.
sec<-«(.-y,
II.
sin^~"aa;cos^~'> — .
12.
sec*-" a;' tan*"
Note. — The student will sometimes find it convenient to invert the
function before differentiation, as we have done in deducing the differen-
tial of sin*-') X.
13. We have, by comparing the above differentials,
<?(sin~^ y + COS"' y) = 0;
d{ta,n~^y + cot"' y) = 0;
a!(sec- ' y -\- csc~ ^ y) = 0.
Show how these results follow immediately from the defini-
tion of complementary functions in trigonometry, combined
with the theorem of § 24 that the differential of a constant
quantity is zero.
33. Logarithmic Differentiation. In the case of products
and exponential functions, it will often be found that the dif-
ferential is most easily derived by differentiating the logarithm
of the function. The process is then called logarithmic dif-
ferentiation.
Example 1. Find -,- when y — x^.
dx ^
We have log y = mx log x;
DIFFERENTIATION OF EXPLICIT FXINOTIONS. 47
~ =m log X dx-\- mdx;
— -i = y{j^i log X -\-m) = mx^'^il -\- log x),
(iX
-, „ sin™ X
Example 2. y = — .
■^ cos" X
We have log y — m log sin x — n log cos a;;
fZy _ in cos a; »t sin x^
ydx sin a; cos x '
dy sin""-' a;. , , . , .
—■ = „ , , (OT cos X 4- 11 sm a;).
(?a; cos" + ia;^ ' '
MISCKLLANEOUS EXERCISES IK DIFFERENTIATION.
Find the derivatives of the following functions with re-
spect to x:
1. y := X log X. Ans. -^ = 1 + log x.
2. 2/ = log tan X. Ans.
dx
dy _ 2
dx sin Hx'
x. y — log cot X. Ans. ~ — : — -—.
^ ^ dx sni2x
X . dy "'
|/(a= - x')' dx {a' - xy
dy _ nx"~
d^~"(r+F}
dy _ 4
_ a;" dy _ ?ia;"~'
^- ^- (1 + a;)"' ^*" ^ " (Th^SJ^^i"
•'«'" + « aa; (e'"+e "')
7. y = log {e- + e-% Ans. ^ = ^-^^^.
8. y=. log tan (J + I). Ans.^=^.
X , dy e'"(l — x) — 1
9- y = ^^r ^'''- ± = (f^vr-
lo „- 4/(l+^)+l/(l- ^)j,,„ f^_ 1
^ ~ 4/(1+:^)- V(l-a;) ^a: ~ a; ^(1 - a;")"
-x'
48 TBE BIFFEBENTIAL CALCULUS.
12. 2/ = tan « -. Ans. % = - '^^^ log a-a\
dx X °
13. y^af^ Ans. j- = 7f{l + log a;).
. ,^ s , dil COS (lofiT a-)
14. y = sm (log x). Ans. -j- = '^ " ■ ' .
, , a; i dy
15. w = tan ' — . Ans. -— — — 7=
" n - x' dx ^/i
,6. 2, = log(^)^-|-tan--..
17. y = log/y/-
- 1 — tan X A dy , , ■ \
18. y — . Ans. -^ = — (cos x + sm x).
sec a; aa;
19. y = log (log a;). ^ws, -^ = —^ .
^ ■^ s \ & / (Za; a; log a;
, 1 — a;" . <fw — 2
y = sm- ' -— — 3. ^Ms. -i- = -— - — 5.
1 + a;" dx 1 -\- X
, , /a cos X — b sin a;
V = log y ; 7^ — -.
a cos a; + 6 sm a;
. dy — ah
Ans. -~- = -5 3 ,a ■ ■ .
«a; a" cos a; — sm' x
22. If « = — , prove the relation ^ - -I - = 0.
dif
23. y =6 -«'*'. Jws. -yi = — 2a'a;y.
. Jtf a;'
aa; 1 — a;
Vl + a;' + a:
Vl + x' — X
A dy 1
20. y
21.
DIFFBBENTIATION OF EXPLIOIT FUNCTIONS. 49
_ 1 1
/|«o '^y ~ ^(^ + «) + n{a + x)
25. y = (a' 4- x') tan~' -. Ans. -— = 3a;tan-> — |- a.
26. y = |/^^-±|y Ans.
^y _
dx (1 — x) \/\ — x"'
11 l'^ \ A ^y 3
^MS. -r- = sin -^.-t: -)- - —
dx Vl-x'
28. y = X sin~' a;.
29. y = tan X tan~ * x.
. dy J . 1 , tan x
Ans. -~ = see" x tan ~'a; + - — ; — ■„-.
ax 1 + a;
30. y = sin W2;(sin x)"
dy
31. y =
Ans. -~ = n (sin a;) "~'sin (» + l)x.
(sin Ma;)""
(cos mxY'
. dy ?nw (sin wa;)""' COB (wia; — wa;)
(cos mx) " + *
32. y = e
Ans.
dx
cos rx
dy _
<?a;
_ p-a-'x'^
e - ""*" (Sa'a; cos rx -\- r sin ra;).
33. y = log-
rt + S tan ]
a — h tan j
'•^MS.
dy
dx
ab
2 2**' Tt • l"*^
34. y = a;-
35. y = sm-
4^'
>.«» ^-^ _ a:^(l - lo g x)
A dy ^
dx
1/1 - 3a; ■
36. y = tan~ ' {n tan x). Ans. -^ =
dx cos' a; + w' sin' a;'
60 THE BIFFMBENTIAL CALCULUS.
dy
37. « = sec~*--7-j jr. Ans. -^= ,. „ ^.
•" ^ y'(ffl' — z') dz y(a — z')
38. y = (a; + a) tan- ' ^|/^ - V(aa;)).
^«s. ^ = tan-'i/^.
dz y a
39. y = sin- ' y(sm z). Ans. -^ — \ |/(1 + cosec z).
4°. y = tan- ' ^^-^. ^^s- ^ = 1+^..
, b + a cos z , dy — ^{a' — V)
41. V = sm- ' — ^^^ . Ans. -f- = -— '-.
^ a-\-ooo&z dz a + cos a;
.a*" - 1 , dy 2nz"-'-
43. y = cos- ^ ^;p^-. Ans. ^ = - ^;,-p-^.
43. 2^ = sec '5-r— 1- ^ws.
22;" - 1" (Za; 4/(1 - a;')'
44. y=tan-^J^i^.4«.. |- = ^^ji_,-.
33. Derivatives with Eespect to the Time. — Velocities. If
we have a quantity which varies with the time, so as to have a
definite value at each moment, but to change its value con-
tinuously from one moment to another, that quantity is, by
definition, a function of the time. "We now have the defini-
tion:
If we have a quantity <p, expressed as a function of the .
time = t, the derivative, -jr, is the velocity of increase,
or rate of variation of <f> at any moment.
This is properly a definition of the word velocity; but it
may be assumed that the student has already so clear a con-
ception of what a velocity is, that he needs only to study the
identity of this conception with that of a derivative relatively
to t, which he can do by the illustration of § 19.
The student is recommended to draw a diagram to rep-
resent the problem whenever he can do so.
DIFFEBBNTIATION OF EXPLICIT FUNCTIONS. 51
EXERCISES.
1. It is found that if a body fall in a vacuum under the in-
fluence of a constant force of gravity, the distances through
which it falls in the first, second, third, fourth, etc., second
of time are proportional to the numbers of the arithmetical
progression
1, 3, 5, 7, etc.,
or, putting a for the fall during the first second, the total fall
will be
a -\- Za -\- ba -{■ la -\- etc.,
continued to as many terms as there are seconds. It is now
required to find, by summing t terms of this progression, how
far the body will fall in t seconds, and then to express its
velocity in terms of t, and thus show that the velocity is
proportional to the time.
, Ans. (in part). The total distance fallen in t seconds will be a^.
The velocity at the end of t seconds will be 2at
2. The above motion being called uniformly accelerated,
prove this theorem: If a body fall from a state of rest with
a uniformly accelerated velocity during any time r, and if the
acceleration then ceases, and the body continue with the uni-
form velocity then acquired, it will, during the next interval
r, fall through double the distance it did during the first
interval.
Find (1) how far the body falls in r seconds; (2) its velocity at the end
of that time; (3) how far, with that velocity, it would fall in another
interval of r seconds; then show that (3) = 2 X (1).
3. The radius of a circle increases uniformly at the rate of
m feet per second. At what rate per second will the area be
increasing when the radius is equal to r feet ?
Find (1) the expression for the value of the radius r at the end of t
seconds, and (2) the area of the circle at that time. Differentiate this
area, and then substitute for t its value in terms of r. Note that {t= — ).
We shall thus have %itmr for the velocity of increase of area.
52 THE DH'FERENTIAL CALCULUS.
4. A body moves along the straight line whose equation is
a; — 2y =
with a uniform Telocity of n feet per second. At what rate
do its abscissa and ordinate respectively increase ?
I-
5. A man starts from a point h feet south of his door, and
walks east at the rate of c feet per second. At what rate is
he receding from his door at the end of t seconds?
Ans. If we put m = his distance from his door, we shall
have
du _ cH
dt^ u'
6, A stone is dropped from a point i feet distant in a hori-
zontal line from the top of a flag-staff 9a feet high. At
what rate is it receding from the top of the flag-staff (1) after
it has dropped t seconds, and (3) when it reaches the ground,
assuming the same law of falling as in Ex. 1?
At the end of t seconds the square of the distance from the top of the
flagstaff — m' = 6' -|- a'f . On reaching the ground we should have
du 54a«
7. The sides of a rectangle grow unifoi-mly, both starting
from zero, and the one being continually double the other.
Assuming one to grow at the rate of m feet and the other 27W
feet per second, how fast will the area be growing at the end
of 1, 2, 10 and t seconds? How fast, when one side is 4 and
the other 8 feet ?
8. The sides of an equilateral triangle increase at the rate
of 2 feet per second. At what rate is the area increasing
when each side is 8 feet long ?
Note that the area of the triangle whose sides = « is ^ * .
DIFPMRBNTIATION OF EXPLIGIT FUN0TI0N8. 53
9. A man walks round a lamp, 30 feet from it, keeping
the distance with a uniform motion, making one circuit per
minute. Find an expression for the rate at which his shadow
travels on a wall distant 40 feet from the lamp.
10. The hypothenuse of a right triangle is of the constant
length of 10 feet, but slides along the sides at pleasure. If,
starting from a moment when the hypothenuse is lying on
the base, the end at the right angle is gradually raised up at
the uniform rate of 1 foot per second, find an expression for
the rate at which the other end is sliding along the base at
the end of t seconds, and explain the imaginary result when
t> 10.
11. Two men start from the same point, the one going
north at the rate of 3 miles an hour, the other north-east 5
miles an hour. Find the rate at which they recede from each
other.
12. A body slides down a plane inclined at an angle of 30°
to the horizon, at such a rate that it has slid M' feet at the
end of t seconds. At what rates is it approaching the ground
(1) at the end of t seconds, and (3) after haying slid 75 feet ?
13. A line revolves around the point [a, h) in the plane of
a system of rectangular co-ordinate axes, making one revolu-
tion per second. Express the velocity with which its intersec-
tion with each axis moves along that axis,- in terms of a, the
varying angle which the line makes with the axis of X.
. dx 3J7r dy 2a7r
Ans. — . J —
dt sin" Of' dt cos" a
14. A ship sailing east 6 miles an hour sights another ship
7 miles ahead sailing south 8 miles an hour. Find the rate
at which the ships will be approaching or receding from each
other at the end of 30, 30, 60 and 90 minutes, and at the
end of t hours.
64 THE DIFFERENTIAL CALCULUS
CHAPTER V.
FUNCTIONS OF SEVERAL VARIABLES AND
IMPLICIT FUNCTIONS.
34. Def. A partial differential of a function of sev-
eral variables is a diflEerential formed by supposing one of the
variables to change while all the others remain constant.
The total differential of a function is its differential
when all the variables which enter into it are supposed to
change.
A partial derivative of a function with respect to a
quantity is its derivative formed by supposing that quantity
to change while all the others remain constant.
Eemaek. The adjective partial may be omitted when the
several variables are entirely independent.
Example. Let us have the function
M = a^Xy + «) + y«- (a)
Differentiating it with respect to a; as if y and z were con-
stant, the result will be
du = 2x{y -f z)dz, (b)
which is the partial differential with respect to x. Also,
is the partial derivative with respect to x.
In the same way, supposing y alone to vary, we shall have
du = {x' + z)dy, (c)
(£)-•+-
PARTIAL DERIVATIVES. 55
which are the partial differential and derivative with respect
to y. For the partial differential and derivative with respect
to z we have
du — (q? + y)dz; (d)
(5) = «■+»■
Notation of Partial Derivatives. 1. A partial derivative
is sometimes enclosed in parentheses, as we have done above,
to distinguish it from a total derivative (to be hereafter de-
fined). But in most cases no such distinctive notation is
necessary.
3. In forming partial derivatives the student is recom-
mended to use the form
Dm instead of t— ,
dx
because of its simplicity. It is called tJie D^ of u. The equa-
tions following (i), (c) and {d) would then be written:
D^u = 2x{y-\-z);
DyU = a;' + «;
D^ti = x' -\- y.
EXERCISES.
Find the derivatives of the following functions with respect
to X, y and 2:
1. V =x' ~ xy -\- y".
Alls. D^v = 2x — y; DyV = — x -\- 2y; D^v = 0.
2. w = x' -j- sj'y + xz. 3. w = x'y'z*.
4. u =x log y -\-y\ogx. S- u = {x -\- y -\- z)'.
6. u — \/{x + my). 7. u = (x -\-2y + 3z)i.
Note. In forms like the last three, begin by taking the
total differential, thus:
du = i(x + 2y + 3zy*d- {x + 2y + 3z)
= i(» + ^y + 3z)"* {dx + 2dy + ddz).
56
THE DIFFERENTIAL CALCULUS.
Then, supposing x alone to vary, Dx^l =
supposing y alone to vary.
2{x+2y+3zf
D«u — 7.
{x + 2y+3zy
supposing z alone to vary, B^u —
2{x+2y+dzf'
8. w = {x -\- y -\- «)". 9.
10. w = cos {mx -\- y). 1 1.
12. ?; = tan (a; — y). 13.
14. V = cos' {ax -\- bz). 15.
16. u = xe^ -\- ye". 17.
w = (a;' + y^ + «")».
w = sin (a; + 2y + 82).
V = sec (w?a; + nz).
w
x" + y'",
18. ?« = sin {x-\-y) cos (a;— «/). 19- u — xsm.y — y sin a;.
35. Fundamental Theoeem. The total differential of
a function of several variables, all of wJiose derivatives are
continuous, is equal to the sum of its partial differentials.
As an example of the meaning of this theorem, take the
example of the preceding article, where we have found three
separate differentials of u, namely, (b), (c) and (d). The
theorem asserts that when x, y and z all three vary, the re-
sulting differential of u will be the sum of these partial differ-
entials, namfely,
du = 2x{y + z)dx -{- (a;" + z)dy -\- {x' -\- y)dz.
To show the truth of the theorem, let us first consider any
function of two variables, x and y.
u = (p{x, y).
(1)
Let us now assign to x an increment /dx, while y remains
unchanged, and let us call u' the new value of it, and A^u the
resulting increment of u. We shall then have
«' = 0(a; + Ax, y);
A^u = <j){x + Ax, y) — t}){x, y).
(3)
TOTAL DIFFERENTIALS. 57
In the same way, if x retains its value while y receives the
increment Ay, and if we call AyU the corresponding incre-
ment of u, we have
AyU = <p{x, y-\-Ay)- (p{x, y). (3)
When Ax and Ay become infinitesimal, these increments
(2) and (3) become the partial differentials with respect to x
and y.
Now, to get the total increment of u, we must suppose both
X and y to receive their increments. That is, instead of giv-
ing y in (1) its increment Ay, we must assign this increment
in (3). Then for the increment of u we shall have, instead
of (3), the result
Ay = (t){x + Ax,y + Ay) - 4>{x + Ax, y). (4)
Note that (3) and (4) differ only in this: that (3) gives the
value of AyU iefore x has received its increment, while (4)
gives AyU after x has received its increment, and is therefore
the rigorous expression for the increment of u due to Ay.
Now, what the theorem asserts is that, when the increments
become infinitesimal, the ratio of Ayu' to AyU approaches
unity as its limit, so that we may use (3) instead of (4). To
show this, let us put
^■<--rt-(s)-
Then, supposing Ay to become infinitesimal, and putting dyio
for that part of the differential of u arising from dy, we shall
have, from (3) and (4),
dyU = cj)'{x, y)dy; (3')
dyu' - <t)'{x + Ax, y)dy. (4')
When Ax approaches zero as its limit, ip'{x + Ax, y) must
approach the limit (p^x, y), unless there is a discontinuity in
68 TBB DlPmnENTtAL CALCULUS.
the function 4>', which case is excluded by hypothesis. Thus,
using (3') for (4')j we have
Total differential of w = du = f-r- \dx + <j>'{x, y)dy
The same reasoning may be extended to the successive cases
of 3, 4, . . . n variables.
The following are examples of finding some differential al-
ready considered in Chap. IV., by this more general process.
1. To differentiate u = xy.
du
dx
= «/;
du
— =x.
dy
Total differential.
du-
= ydx
■ + xdy.
2.
X
u = — = xy-
■ 1
du
di=y
-I .
3
du
dy
■ xy ' %;
du = y~ ^dx — xy ^dy = 5 — -.
3. u = ax -\- bxy -\- cxyz.
^^ = « + ^y + "y^;
du, ,
— - = bx-\- cxz;
dy
du
du = (a + Jy + ctjz)dx + {hx + cxz)dy + cxydz,
as in § 25, Example 1.
DIFFERENTIATION OF IMPLICIT FUNCTIONS. 69
EXERCISES.
Write the total difEerentials of the functions given in the
exercises of § 34.
36. Principles Involved in Partial Differentiation, All
the processes of the present chapter are aimed at the following
object: Any deriyative expression, such as
du _
di' °'^^'''
presupposes (1) that we hare the quantity u given, really or
ideally, as an explicit function of x, and perhaps of other
quantities; (3) that we are to get the result of differentiating
this function according to the rules of Chap. IV., supposing
all the quantities except x to be constant.
Now, because it is often difficult or impossible to find u as
an explicit function of x, we want rules for finding the values
of D^u, which we could get if we had u given as such a func-
tion of X. For example, we might be able to find the equa-
tion u = <p{x) if we could only solve one or more algebraic
equations. If, for any reason, we will not or cannot solve
these equations, we may still find D^u whenever the equations
would suffice to give u as a function of x if we only did
solve them. The following articles show how this is done in
all usual cases.
37. Differentiation of Implicit Functions. Let the rela-
tion between y and x be given by an equation of the form
0(a;, y) = 0. (a)
Eepresenting this function of x and y by 0, simply, and
supposing for the moment that x and y are independent
variables, so that need not be zero, we shall have, by the
last section,
deb = -j^dx + -r^y-
^ dx dy ^
60 THE HmVERENTIAL CALCULUS.
But, introducing the condition that equation (a) must be
satisfied, tZ0 must be zero, because x and y must so vary as to
keep constantly zero. We then find, from the last equation,
dz d^ Dy^' ^ '
dy
which is the required form in the case of an implicit function
of one variable.
Cor. If from an equation of the form x =f{y) we want to
derive the value of D^y, we have
^{x, y)=x -f{y) = 0;
<?0 _ d<p _ df(y) _ dx
dx ~ ' dy '^ dy ~ dy'
Hence -^ = -r-.
dx dx
dy
Example. To find D^y from the equation
9^(*' y) = y — ax — li.
Wehave g = - «; f = 1; % = a',
the same result which we should get by differentiating the
equivalent equation y — ax.
Remabk. If -we should reduce the middle member of (1) by clearing
of fractions, the result would be the negative of the correct one. This
illustrates the fact that there is no relation of equality between the two
differentials of each of-the quantities x, y and 0, all that we are concerned
with being the limiting ratios dy -.dx; d4>: dx, and d<t> : dy, which limit-
ing ratios are functions of x and y.
We may, indeed, if we choose, suppose the two ds's equal and thetwo
dy's equal. But in this case the two d(p'a must have opposite algebraic
signs, because their sum, or the total differential of cp, is necessarily zero.
Now, if we change the sign of either of the dcp's, wc shall get a correct
result by a fractional reduction.
DIFFERENTIATION OF IMPLICIT FUNCTIONS. 61
EXERCISES.
Find the values of -—, -^ or -5- from the following equa-
tions:
I. y' — ax = 0. 2. y'' — yx-^-x' = 0.
3. x' + ixz + 2' = 0. 4. u{a-x)-\-u\b + x) = 0.
S- log a; + log y = c. 6. log (a;+z/) + log {x-y) = c.
7. sin a; + sin y = c. 8. sin ax — sin % = e.
9. M 4" e sin ?« = X. 10. x {1 — e cos 2) = a.
38. Implicit Functions of Several Variables. The pre-
ceding process may be extended to the case of an implicit
function of any number of variables in a way which the
following example will make clear.
Let u be expressed as a function of x, y and z by the
equation
u' + xu' + {x' + y')u + x' + y' + z' = 0.
Since this expression is constantly zero, its total differential
is zero. Forming this total differential, we have
{3u' + 2xu + x' + y')du + («' + 2ux + 3x')dx
+ {2uy + 3y')dy + 3z'dz = 0.
By § 34 we obtain the derivative of u with respect to x by
supposing all the other variables constant; that is, by putting
dy = 0, dz = 0, and so with y and z. Hence
dtt _ u' -\- 2ux -4- Sa;'
-^ = Dm =
dx " 3u' + 2ux + a;" + y"
du _ _ 2m/ + 3y^
--_V^u- - 3jj= _^ 2ux + a;' -f y"
du _ 3z^
-rr- = Dm
dz ' 3u' + 2ux + a;" + y''
62
THE DIFFERENTIAL CALCULUS.
EXERCISES.
Find the derivatiyes of u, v or r with respect to x, y and e
from the following equations:
1. xu' + ifu' + z'u — x'yz.
2. a cos {x — ti) -\-b sin {x -\- u) = y.
4. r-
a! + »
I ij-x — v ^ jje_
6. e"" cos a; + e"" cos y = e".
8. V° + 'HVX cos -f k" r= 5",
3. u" + zt" = ?«'.
5. V log a; + 2 log V = y.
7. ?t" — Swa; cos z -f- a;' = a',
39. Case of Implicit Functions expressed by Simvlta-
neotis Equations. If we have two equations between more
than two variables, such as
F^u, V, X, y, etc.) = 0, F^{u, v, x, y, etc.) = 0,
then, if values of all but two of these variables are given, we
may, by algebraic methods, determine the values of the two
which remain. We may therefore regard these two as func-
tions of the others, the partial derivatives of which admit of
being found.
In general, suppose that we have n independent variables,
a;,, x^. . . ar„, and ?« other quantities, u^, u^ . . . Un, connected
with the former by m equations of the form
F^u^, M, . . . M„, a;,, a;, . . . a;„)
i^,(M., M,
. a;„) =
(«)
i^m(w„ M, . , . Mm, «„ a;, . . . a;„) = 0.
By solving these m equations (were we able to do so) we
should obtain the m u'& in terms of the n a;'s in the form
Ml = 0,(a;., a;, . . . a;„);
Mr.= 0m(a;„a:, . . . a;„); _
(6)
DIFFERENTIATION OF IMPLICIT FTJNGTIONB. 63
and by differentiating these equations (5) we should find the
mn Talues of the deriTatives -j-'; -=-'; . , . -r^; etc.
'Now, the problem is to find these same derivatiyes from (a)
without solring (a).
The method of doing this is to form the complete differen-
tial of each of the given equations (a), and then to solre the
equations thus obtained with respect to clu^, du^, etc.
The results of the differentiation may, by transposition, be
written in the form
dF, , , dF,^ , , dF, ^ dF^ , ,
^r^ du, 4- -^ du, + . . . + t— du^ = — ^ dx, — etc. :
du^ ' du, ' du„^ dx^ '
dF, , , dF,^ , , dF,^ Fd, , ,
dF^ , , dF^ _, , , dF^ -. dF^ , ,
_^^,.+__,?^, + . . . +_^«„:=: __^.,_ etc.
By solving these m equations for the m. unknown quantities
du^, du, . . . dUm, we shall have results of the form
du^ = M^dx^ + M,dx, + • ■ • + ^ndx„;
du, =? W^dx^ + JV,dx, + . . . + NJLx^;
etc. etc. etc. etc.;
where M^, iV„ etc., represent the functions of u^. . . u^,
x^ . . . x„, which are formed in solving the equations.
We then have for the partial derivatives
^i_jf. ^^-M- etc
Example. Prom the equations
rcosd = x,) ^^,j
r sin 6* = y, )
it is required to find the derivatives of r and 6 with respect
to X and y.
64 THM DIFFERENTIAL CALCULUS.
By differentiation we obtain
cos ddr — r sin 6d6 = dx;
sin 6dr + r cos 6d6 = dy.
Multiplying the first equation by cos 6 and the second by
sin 6, and adding, we eliminate dO. Multiplying the first by
— sin 6 and the second by cos 6, and adding, we eliminate dr.
The resulting equations are
dr = cos ddx + sin ddy;
rdd = cos ddy — sin ddx.
Hence, as in the last section,
(I) = ^°^ ^' (S) = '^ ^'
ldff\ _ _ sva_e_ lde\ _ cosg
Wk/ ~ r ' \f^y/ ~ r
EXERCISES.
1. From the equations
>• sin ^ = a; — y,
r cos 6 = x-\- y,
find the deriyatives of r and ^ with respect to x and y.
2. From the equations •
Me" = r cos ^,
?(e~"= r sin ^,
find the derivatives of u and v with respect to r and 6.
Ans. [-^) = Ue" sin (? + e - " cos (9) ;
^(e-''sin (9 + 6" cos (9).
(-) =
\de}
FUNCTIONS OV FUN0TI0N8. 65
3. From the equations
u" -\- ru = x' -\- y',
m" — ru = xy,
And the derivatives of r and u with respect to x and y.
4. From the equations 5. From
a;" + «/' + «' - '^xyz = 0, m' - 3«;« cos 6 -^r z'' - a',
x-\-y -\- z = a, id' -\- 2uz cos 6 + z" — b',
„ ■, dz J dz „ . du du dw dw
find J— and -j-. find -3-; -=^; -r~; -ttt.
dx dy dz' dd ' dz d6
40. Functmis of Functions. Let us have an equation of
the form
u-f{<P, t, 6, etc.); {a)
where <p, ip, d, etc., are all functions of x, admitting of being
expressed in the form
<P^fA^); i^=.a^); 6=f,{x); etc. {b)
If any definite value be assigned to x, the values of cp, f,
6, etc., will be determined by [b). By substituting these val-
ues in (rt), u will also be determined. Hence the equations
(«) and {b) determine u as a function of x.
By substituting in (a) for 0, tp, B, etc., their algebraic
expressions /,(a;), fi{x), etc., we shall have u as an explicit
function of x, and can hence find its derivative with respect
to X. But what we want to do is to find an expression for
this derivative without making this substitution.
By differentiating (a) we have
^„ = ^^0_^grf^ + ^^^ + etc.
By differentiating (^),
d(t> = -^dx\ dtp — -^dx; d6 = -^f-dx; etc.
dx dx dx
66 THE DIFFEBENTIAL CALCULUS.
By substituting these values in tlie last equation and divid-
ing by dx, we have
du _ du d(p die dij) du ^^ _, j. /-,\
dx d<p dx d^ dx dQ dx ' ^ '
The significance of this equation is this: a change in z
changes u in as many ways as there are functions (p, ip, 6, etc.
-5--r -^dx is the change in u through ^;
-jj -T-dx is the change in u through ip;
etc. etc.
The total differential is the sum of all these separate
infinitesimal changes, and the derivative is the quotient of
this total differential- by dx.
EXERCISES.
1. Find -5— from the equations
u ^ a sin {inv -f- w) + 5 sin {mv — w);
«; =: c -f- ii^'> w = c ~ nx.
We find -=- = am cos (mv -\-w)-{-bm cos (mv — w); -,— = n;
dv ' dx
du , , ^ I , s dw
-— = a cos (mv + «c) — J cos (mv — w); —- =: — n:
dw " dx
Whence, by the general formula,
-— = an{m — 1) cos (jnv -\-w)-\- bn^m -\- 1) cos {rrm — w).
2. Find ^— from
dx
w = e* + e*;
•3. Find -Y- from
v' + v<p + ip'' = a;
= OT(a + y); f = ny.
FUNCTIONS OF FUNCTIONS. 67
4. Find -^- from
r cos X — r sm x = a — y,
X = inz-\-h; y = 00s nz.
S. Find -=- from
^ dz
r' + ar' + 2/V + 0= = 0;
a;" + az = 0; y' -{■ az' = 0; = mz.
41. The foregoing theory applies equally to the case in
which the function is one of two or more variables, some of
which are functions of the others. For example, if
u = (t>{x, z), {a)
then, whatever be the relation between x and z, we shall
always have, for the complete differential of u,
au=[^)dx + (^)dz.
Suppose that x is itself a function of z. We then have
, dx ^
dx = ^ dz.
dz
By substitution in the first equation we have
du —
Wa; Idu
Idz + Vdz
dz;
du _ Idu \ dx Idu \ . .
''"dz~\dxJ dz'^Vdz)' ^*^
The two values of -=- which enter into this equation are
different quantities. A change in z produces a change in u
in two ways: first, directly, through the change in z as it
appears in (j); second, indirectly, by changing the values of
X in (a). The first change depends upon (-j-j in the second
68 THE DIFFERENTIAL CALCULUS.
member of (5); the second uponl-^j --^; while the first mem-
ber of {b) expresses the total change.
It is in distinguishing the two values of a derivative thus
obtained that the terms 2}(t'>'Hal derivative and total derivative
become necessary. If we have a function of the form
w =f{^> y,w . . . z),
in which any or all of the quantities x, y, w, etc., may be
functions of z, then the partial derivative of u with respect
to z means the derivative when we take no account of the
variations of x, y, w, etc.; and the total derivative, with
respect to z, is the derivative when all these variations are
taken into account.
In such cases the partial derivative has to be distinguished
by being enclosed in parentheses (§ 34). This is why the last
equation is written
du _ fdu\ ldu\ dx
dz ~ \dzJ \dx J dz'
43. Extension of the Principle. The principle involved
in the preceding discussion may be extended to the case of
any number of independent variables and any number of
functions. If we have
r = 4>(u, V, w . . . X, y, z . . .),
in which x, y, z, etc., are the independent variables, while
u, V, w, etc., are functions of these variables, we shall have
*=(!?)*' + ©'"+■■• + ©*+*■
Then, since u, v, lu, etc., are functions of x, y, z, etc., we
have
du = -j—dx + T— </t/ + etc. ;
dx dy
dv = -^dx + -^dy 4- etc.
dx dy •'
FUNCTIONS OF FUNCTIONS. 69
By substituting these values in the preceding equation we
find*
dx
~~ {\dxj \duj dx \dv J dx
•
r(d<p\ /d(p\ du fd<p\ dv
"*" LW j "^ [duj dy '^\dvl dy'^ "
•
+
e, writing r for cp, its equivalent,
cZr /«?r\ 1 dr\du ldr\dv
dx ~ [dx] \dujdx Kdvjdx "'
t
etc. etc. etc. etc.
EXERCISES.
The independent variables r and d being connected with x
and 2/ by the equations
a; = r cos d,
y =^ r sin 6,
it is required to find the derivatives of the following functions
of X, y, r and d with respect to r and 0. We call each of the
functions u.
I. u — r" + 2xy cos 36*.
Here we have
— - = 2w COS 28; -— = 2x COS 38;
dx . dy
-,— = cos 8; -f- = sme;
dr dr
dx . . dy .
— „- = — r sm 8 = — «; -^/r = r cos 8 = a;.
flO BO
* Here, when we use the symhol (p instead of r, there is really no
need of enclosing the partial derivatives in parentheses. We have done
It only for the convenience of the student.
70 THE DIFFERENTIAL CALCULUS.
_ du ldv\ , du dm , du dv
Hence -r- = (3-I + -j- 3- + -j- -r-
dr \drf dx dr dy dir
= 2r4-33rcose coa29 + 2a:sm9 cos 29
= 2r(l + cos 29 sin 29) = r{2 + sin 49);
and, in the same way,
~ = 2r^ cos 49.
We might have got the same result, and that more simply, hy sub-
stituting for X and y in the given equation their values in terms of r and
6. But in the case of implicit functions this substitution cannot be
made; it is therefore necessary to be familiar with the above method.
2.
u^\ + '' J cos 2ft
r a
3-
'' = x' + y' r''-
4.
u^r' - {x- yy.
S-
1
~ xsmllO -\- y cos 2^*
6.
1 1
U = -^ — : ^^.
X COS 20 y sin 2^*
7. u = r' -\-x' — y\
Let V and w be given as implicit functions of p and d by
the equations
v' + w= = 2p sin ft f ^"'
It is required to find the total derivatives of the following
functions with respect to p and 6 respectively:
8. ti = v' -\- w' — p'. g. u = v' — 2vw cos + w'.
10. V, — — . II. w = (y + w) sm ft
12. u ^= {v — to) cos ft
13. M = iw" — f " -)- 2(?<; "l" v)p cos ft
PARTIAL DERIYATIVES. 71
From the pair of equations (a) we find
do
_ « dw _ w ^
~2p'' dp~ %p'
= i« cot 9; -^- z^w cot
which values are to be substituted in the symbolic partial derivatives of il.
43. Reniarhs on the Nomenclature of Partial Derivatives.
There is much diversity among mathematicians in the no-
menclature pertaining to this subject. Thus, the term " par-
tial derivative" is sometimes extended to all cases of a deriva-
tive of a function of several variables, with respect to any one
of those variables, though there is then nothing to distinguish
it from a total derivative.
Again, Jacobi and other German writers put the total deri-
vatives in parentheses and omit the latter from the partial
ones, thus reversing the above notation.
If we have to express the derivative of <p{x, y, z, etc.) with
respect to z, the English writers commonly use the symbol
-T- in order to avoid writing a cumbrous fraction. We thus
have such forms as
dxW '^ V '^ c'J'
each of which means the derivative of the expression in paren-
theses with respect to x, and which the student can use at
pleasure.
44. Dependence of the Derivative upoti the Form of the
Function. Let x and y be two variables entirely independent
of each other, and
u = (p{x, y) {a)
a function of these variables. Without making any change
in u or x, let us introduce, instead of y, another independent
72
THE DIFFERENTIAL OALGULUS.
variable, z, supposed to be a function of x and y. Then, after
making the substitution, -we shall have a result of the form
u = F{x, z). (b)
Now, it is to be noted that although both zi and x have the
same meaning in {b) as in («), the value of -^ will be differ-
ent in the two cases. The reason is that in («) y is supposed
constant when we diiferentiate with respect to x, while in (J)
it is z which is supposed constant.
Analytic Illustration. Let us have
u = ax' -\- by'.
This gives
du -
T— = %ax.
ax
{0)
Let us now substitute for y another quantity, z, determined
by the equation
z = y -\-x or y = z — X.
We then have u = ax" -\- b{z — xY;
— - = 2ax -\- 2b(x — z);
which is different from (c).
Our general conclusion is: The partial derivative of one
variable with respect to another depends not only upon the re-
lation of those two variables, but upon, their relations to the
variables which we sup-
pose constant in differen-
tiating.
Geometrical Illustra-
tion. Let r and be the
polar co-ordinates of a
point P, and x and y its
rectangular co-ordinates, q
Then
X
•R
y
1
3fr—y
Q
\ ^y
•'^S
y^ y'
/ ,-'
J^ #'
/V^'
/ /
/>
y^'
y>^
o
hX
x:
Fio. 9.
r cos 6;
y = >• sin d;
r' = a' + f.
{d)
PARTIAL DEBIVATIVEB. 73
Eegarding r as a function of x and y, we haye
But we may equally express r as a function of x and 6, thus:
r = a; sec 6. (/)
We then have -^ = sec ft (gr)
Referring to the figure, it will be seen that we derive (e)
from {d) by supposing x to vary while y remains constant;
that is, by giving the point P an infinitesimal motion along
the line PQ || to OX. In this case it is plain that the incre-
ment of r (SQ) is less than that of x. But in deriving (g)
from (/) we suppose x to vary while 6 remains constant.
This carries the point P along the straight line OPR; and
now it is evident that the resulting increment of r {PR) is
greater than that of x.
74
THE DIFFERENTIAL CALGULUB.
CHAPTER VI.
DERIVATIVES OF HIGHER ORDERS.
45. If we have given a function of x,
y = <>(x),
we may, by difEerentiation, find a value of ■—. This value
will, in general, be another function of x, which we may call
4>'{x). Thus we shall have
Ifow, this function 0' may itself be differentiated. If we
call its derived function (f>",
we shall have
4-
ax
dx
dx
= 0"(^). («)
Let us examine the geo-
metrical meaning of this
equation, by plotting the
curve representing the origi-
nal equation y = <p (x).
Let X, x' and x" be three
equidistant values of the ab-~
scissa, so that the increments
x' — X and x" — x' = Ax are
equal. Let P, Q and R be
the corresponding points of the curve,
be the three corresponding values of y.
B
M
AX
Xo
X2
Fia. 10.
Let y, y' and y''
DEBIVA TIVES OF EiaHEB 0BBEB8. 75
Then we may put
Ay = y' -y = MQ,
A'y = y" -y' = NR,
as the two corresponding increments of y.
It is evident that these increments will not, in general,
be equal; in fact, that they can be equal only when the three
points of the curve are in the same straight line. If D is the
point in which the line PQ meets the ordinate of R, then
DR will be the difference between the two values of Ay, so
that we shall have
BR = A'y — Ay ■= increment of Ay.
Hence, again using the sign A to mark an increment, we
shall have
BR = A Ay = A'y, (b)
in which the exponent does not indicate a square, but merely
the repetition of the symbol A.
Thboeem I. When Ax becomes infinitesimal, A'y becomes
an infinitesimal of the second order.
For, if B be the point in which PQ produced cuts the
ordinate X,i2, we shall have, in the triangle QRB,
nB=QB'^^ = A'y. (b)
sm QRB " ^ '
When Ax becomes an infinitesimal of the first order, so do
both QB and the angle RQD, but the angle QRB will remain
finite, because it will approach the angle QBN as its limit.
Hence the expression will contain as a factor the product of
two infinitesimals of the first order, and so will be an infini-
tesimal of the second order.
Since both the quantities QB and RQB depend upon Ax,
we conclude that the ratio
A^
Ax'
may remain finite when Ax becomes infinitesimal. In fact.
76 THE DIFFERENTIAL 0ALCULU8.
from the way we have formed these quantities, we have
hm. ^ = lua.-^ = -^ = 0'».
Hence —
Theorem II. If we take two equal consecutive infinitesimal
incremsnts, = dx, of the independent variable, then —
1. The difference between the corresponding infinitesimal
increments of the function divided by dx' will approach a
certain limit, -
2. This limit is the derivative of the derivative of the
function.
Def. The derivative of the derivative is called the second
derivative.
The derivative of the second derivative is called the third
derivative, and so on indefinitely.
Notation. The successive derivatives of y with respect to
X are written
dy . ^. ^. . .
dx' dx" dx" ®°-'
or D^y; D^y; D^^'y; etc.
46. Derivatives of any Order. The results we have
reached in the last article may he expressed thus: If we have
an equation
y = <P{x),
the first derivative is given by the equation
Then, by differentiating this equation, we have, by the last
theorem,
dx d^y ,,,, .
DEBIVATIVES OF EIOBEE ORDERS. 11
Again, taking the derivative, we have
and we may continue the process indefinitely.
EXERCISES AND EXAMPLES.
1. To find the successive derivatives of ax'.
and all the higher derivatives will vanish.
Form the derivatives to the third, fourth or ?tth order of —
2. ax\ 3. hx~^. 4. {a-\-xY.
5. {a — xy. 6. (a-j-a;)~l 7. (a — x)~^
8. [a" + xy. 9. -Zd'x' + x\
10. a-\-hx -\- ex' -\- hx' + kx\
11. 1 + a; + a" + a;' + »* + a;' + . . . + a;».
12. 1 - a; + a;= - a;' + a;' - a;' + . . . H- (- l)"a;"-
13 n
13. a?. 14. a;2. 15. (rt -(- a;)" 16. (a + a;)^.
17. If «/ = e', find Z*/;;/ = a''(log a)".
18. From y = m(f, find the «th derivative.
19. From y = me*'' show that D^y = A"«/.
Find the first three derivatives of the expressions:
20. 2^ 21. aof. 22. x'^"-
23. log X. 24. log (a + x). 25. »7i log X.
26. log (ffl — a;). 27. log (a + "*-'^)' ^8. log (a — ma;).
29. Show that if y — sin x, then -j^ = — y.
d'y _ _ tf + ^y _ ^
W ^^^'' (?«" + " ~ <?a;»'
78 THE DIFFERENTIAL CALCULUS.
30. Show that the same equations hold true H y = cos x
or if y = a cos a; + J sin x.
31. Find the law of formation of the successive derivatives
of sin mx and cos mx.
Especially, the {n + 4)th derivative = wth der. X what?
(« + 2)th derivative = wth der. x what?
32. Find the «th derivative of e^".
33. Find three derivatives of e"" sin nz.
34. If M = y", show that ^ = (1 + log y)-^ + ^.
35. Find two derivatives of m = tan z.
3fr. Find two derivatives of m = cos" z.
37. Find two derivatives of m = sec' z.
38. Find two derivatives of m = cos' z — sin' z.
39. Find two derivatives of « = cos 22.
40. Find two derivatives of u = e'"'.
41. Find two derivatives otu = sin'~"a;.
47. Special Forms of Derivatives of Circular and Ex-
ponential Functions. Because
cos x = sin {x -\- ijt) and — sin a; = cos (z -f ^tt),
the derivatives of sin x and cos x may be written in the form
Bx sin X = sin {x -f- ^tt)
and Dx cos x = cos (x -j- ^n).
Hence, the sine and cosine are such functions that their
derivatives are formed ly increasing their argument iy ^rr.
Differentiating by this rule n times in succession, we have
_ . . d" sinx . f , n \
Bx- cos X = -^— = cos [x + ^^j;
results which can be reduced to the forms found in Exercises
29 and 30 preceding.
DMBIVATIVE8 OF HIGHEB 0BDEB8. 79
48. Successive Derivatives of an Implicit Function. If
the relation between y, the function, and x, the independent
variable, is given in the implicit form
f{x, y) = 0,
then, putting u for this expression, we have found the first
derivative to be
du
dy _ dx . ,
dx ~ du' ^ '
dy
The values of both the numerator and denominator of the
second member of this equation will be functions of x and y,
which we may call X^ and Y^. We therefore write
^y=-^. (V)
dx y; ^"^
Differentiating this with respect to x, we shall have
d-'y _ ' dx '^ ' dx
dx' ~ y; ^"^
X^ and Y^ being functions of both x and y, we have (§ 41)
dx \dx I \ dy Idx'
dT\ ^ fdY,\ fdYXdy
dx \ dx I \dy Idx'
Substituting in these equations the values of -^ from (J),
and then substituting the results in (c), we shall have the re-
quired second derivative.
The process may then be repeated indefinitely, and thus
the derivatives of any orders be found.
Example. Find the successive derivatives of y with re-
spect to X from the equation
x' — xy -\- y^ = u = 0,
80 THE DIFFERENTIAL 0ALCULU8.
We have £ = Zx-y; ^-=-x + %y;
dy 2a; -y .
dx X- 2y' ^" '
which is a special case of (a) and (b), and where
X^ =: 2x — y and F^ =: — a; + 2y.
Differentiating the equation {a'), we haye
d'y_('= ^y) dx ^^"^ y'~ dx
dx' (x - 2yy
(x - 2yy
Substituting the value of ~ from (a'), we have
dy ^ (x - 2y) (- 3.y) + 3a:(2a; - y)
dx' {x — 'ZyY
_ 6 (a:' — xy -f y') _ 6m
(a: - %yY ~ {x - 2yy'
EXERCISES.
Find by the above method the first two or three derivatives
of V with respect to x, y or z, from the following equations:
, \ J d'v 2(a -\- v)
1. zv = a(v — z). Ans. -t-=- = -r — ■ — ri.
' dz' (a — zf
2. v'y -(- vy' = a.
3- v' -\- vz -\- y' = b.
4. v{a — xy + v'{b -\-x) = c.
5. log {v + z)+ log (y ~z) = c.
6. sin inv — sin ny = ^.
7. t)(l — a cos «) = h.
8. If M — e sin w = (/, show that
d'u 1 — e
dedg (1 — e cos «)**
DERIVATIVES OF HIGHER ORDERS. 81
49. Leibnitz's Theorem. To find the successive deriva-
tives of a product in terms of the successive derivatives of its
factors.
Let iiv =j9 be the product of two functions of x. By suc-
cessive differentiation we find
dp _ dv du_
dx~ dx dx'
d^p _ d'v du dv d'u
dx' dx" dx dx dx" '
d'p _ d'v du d'v d'u dv d'u
dx" ~ dx' dx dx" dx" dx dx'
So far, the coefficients in the second member are those in
the development of the powers of a binomial. To prove that
this is true for the successive derivatives of every order, we
note that each coefficient in any one equation is the sum of
the corresponding coefficient plus the one to the left of it in
the equation preceding. Now, let us have for any value of n
d'^v d^v , du d^-^v , ^ , ,
the successive coefficients being
1; n; (l); (^^]; etc. (Comp. § 6.)
Then, in the derivative of next higher order the coefficients
will be
'■■'+'■■ ©+» " m^
and, in general.
(?)+fe) »' m-
I'^ + ^r.
That is, J „^i is formed from (a) by writing n -\-l for n.
Hence, if the rule is true for n, it is also true for n-\- 1. But
it is true for w ?= Sj . • , for w = 4, etc, indefinitely.
82 TEE DIFFERISNTIAL CALCULUS.
50. Successive Derivatives with respect to Several Equi-
crescent Variables. Studying the process of § 45, it will be
seen that we supposed the successiye increments of the inde-
pendent variable to be equal to each other, and to remain
equal as they became infinitesimal, while the increments of
the functions were taken as variable. This supposition has
been carried all through the subsequent articles.
Def. A variable whose successive increments are supposed
equal is called an equicrescent variable.
We are now to consider the case of a function of several
equicrescent variables.
If we have a function of two variarbles,
u = (t>{%, y),
the derivative of this function with respect to x will, in
general, be a ftinction of x and y. Let us write
-^ = Mx,y).
Now, we may differentiate this equation with respect to y
-ivith a result of the form
du
-^ = (p^,^{x, y).
Using a notation similar to that already adopted, we rep-
resent the first member of this equation in the form
dxdy'
In the D-notation this is written
In either notation it is called "the second derivative of
u with respect to x and y."
As an example: If we differentiate the function
u = y'' sin (rnx — ny) (a\
BEBIVATIVEa OF HIGHER ORDEBB. 83
with respect to x, and then differentiate the result with
respect to y, we have
DxU = -J— = i^y cos (mx — ny) ;
d'u
D'x,yU — -J = '^my cos (mx — ny) -\- mny" sin {vix — ny).
51. We now have the following fundametftal theorem:
d'^u __ d'li, _
dxdy dydx'
or, in words.
The second derivative of a function with respect to two
equicrescent variables is the same whether we differentiate in
one order or the other.
Let u = <p{x, y) be the given function. Assigning to x
the increment Ax, we have
^ ^ ^{x + Jx, y) - (p{x, y)
Ax Ax ' ^ '
All
In this equation assign to y the increment Ay, and call A—r-
^x
Att
the corresponding increment of --r-. Then the equation will
give
^ + ^^= 0C-g + Ax,y + Ay) - cjy{x, y + A y)
Ax Ax Ax ' '
Subtracting (1) and dividing the difference by Ay, we
have
^^ _ 0(g +Jx,y-\- Ay) - (t>{a:, y J ^ Ay) - ct>{x + Ax. y ) + <;6(a, y )
Ay Ax Ay
The second member of this equation is symmetrical with re-
spect to X and y, and so remains unchanged when we inter-
change these symbols. Hence we have
,Au .Au
Ax _ Ay
Ay ~ Ax
84 THE DIFPBBENTIAL CALCULUS.
for all values of Ax and Ay, and therefore for infinitesimal
values of those Lucrements. Thus
^chi , du
d-^ d--^r-
dx _ ay
dy ~ dz '
or D\,yU = D\,^u,
as was to be proved.
As an example, let us find the second derivative of (a) in
the reverse order. We have
-^ = %y sin (mx — ny) — ny" cos {mx — ny);
ay
d^u
-— -- = %ny cos {rnx — ny) -\- mny^ sin {vix — ny);
the same value as before.
Corollary. 7" he result of talcing any number of succes-
sive derivatives of a function of any number of variables is
independent of the order in ivhich ive perform the differcntia-
iions.
For, by repeated interchanges of tw^o successive differentia-
tions, we can change the whole set of differentiations from
one order to any other order.
If we have I differentiations with respect to x, m with re-
spect to y, n with respect to z, etc., and use the Z>-notation,
we express the result in the form
Here the symbol D^ means DyBy, etc., m times.
In the usual notation the same operation is expressed in
the form
cF + ™ + " + ---0
dofdy'^dz" . . .'
The corollary asserts that, using the Z>-notation, we may
permute at pleasure the symbols DJ, B^, D^, etc., without
changing the result of the differentiations.
DMHIVATIVES OP EI&EER ORDMBS. 85
EXERCISES.
Verify the theorem DJ)yU = DyD^u in the following cases:
I. u = X sin y -\- y sm x. 2. u = x".
3. u = xlogy.
4. u = a sin {x -\- y) — b sin {x — y).
Differentiate each of the following functions once with re-
spect to z, twice with respect to y, and three times with re-
spect to X, in two different orders, and compare the results.
7. xsva. y -\- 2/ sin z -\-z sin x. 8. sin {Ix -\- my + nz).
1
9. If u —
'V
^{x' + f-\- z')
, ,, , d'u , d''u , d'u
show that -_+_, + _, = 0.
53. Notation for Powers of a Differential or Derivative.
Such an expression as du' may he ambiguous unless defined.
It may mean either
Differential of square of u; i.e., diu');
or Square of differential of 11; i.e., {du)'.
To avoid ambiguity, the expression as it stands is alwayn
supposed to have the latter meaning. To express the differ-
ential of the square of u we may write either
d'u' or (?(«"),
of which the first form is the easier to use.
The square of the derivative -p- may be written either
Idu y du*
\dx I do?'
86 THE BIFFEItmNTIAL CALCULUS.
CHAPTER VII.
SPECIAL CASES OF SUCCESSIVE DERIVATIVES.
63. Successive Derivatives of a Power of a DerivativBc
Let us have to differentiate the derivative
■with respect to x.
In such operations the i^-notation will he found most con-
venient.
Applying the rule for differentiating a square, the result is
j/duV du
\dx) _ du dx _ du ^
dx dx dx ~ dx dx"
or, in the i)-notation,
D^iD^u)' = 2D^uDM
In the same way, we find
d-{I)^uY (duy-^d'u ,„ ^„ .„,
SPECIAL CASES OF SUCCESSIVE DERIVATIVES. 87
EXERCISES.
Write the derivatives with respect to x of the following ex-
pressions, y being independent of x when it is written as an
equicrescent variable:
■• (I)" '■ (I)" - (I)'
^- ^©' »• '©■■ '■ 4)"
7- \w ' ^- wj • ^' \^ ■
^°' d«^' ''■ \dxl dy ' \dxl \dyl '
(dyVlduV (duV/d'uy IdhiVld'uV
'3- liJ \dz) ' ''• \di) \d^} • '5- [ay) W-j '
fd'yVfd'yY dudv_dudv fd^uV
54. Derivatives of Functions of Functions. Let us have,
as in § 40,
M=/-W, (1)
where ^ is a given function of x. It is required to find the
successive derivatives of u vrith respect to x. We may evi-
dently reach this result by substituting in (1) for f its ex-
pression in terms of x, and then differentiating the result by
methods already found.
But what we now wish to do is to find expressions for the
successive derivatives without making this substitution. To
do this, assign to x the infinitesimal increment dx. The re-
sulting infinitesimal increment in ip will be
df = ^dx.
88 TaS DIFfERBNTlAL CALOVHTS.
This, again, will give u the increment
du = g#,
or, by substituting for di/^ its value, and passing to the de-
rivative,
du _ du dip ,„,
dz dip dx'
This is a particular case of the result already obtained in
§40. The second member of (2) is a product of two factors.
The first of these factors is formed by differentiating a func-
tion of ip with respect to if); and is therefore another (derived)
function of ip; while the second is, for the same reason, a
function of x.
Differentiating (2) with respect to x by the rule for a prod-
uct, we have*
^du
d'u _dip dip du d'lp . .
dx' ~ dx dx dip dx'' ^ '
N"ow, because -rr is a function of ij), its derivative with re-
dip
spect to X is to be obtained in the same way as that of u.
If we put, for the moment,
«' = !=/•»).
we have, as in (3),
du' _ du' dip _ d^u dip ^
dx ^ dip dx ~ dip' dx '
* The student should note that the expression —: — cannot be put in the
cPu
form — r-;-, because the latter form presupposes that ib and x are two in-
dijidx
dependent variables, which is here not the case. In fact, v, does not con-
tain X except in ip.
SPECIAL OASES OF SUCCESSIVE DEBI7ATIVE8. 89
and hence, by substitution in (3),
d'u _ d'uldipV du d'lp ,...
d^' ~ dip'W) "^ # d^"'' ^ '
which is the required expression for the second derivative.
From this we may form the third and higher derivatives
by again applying the general rule embodied in (2), namely :
Ifipisa function of x, we find the derivative of any func-
tion, u, ofiphy differentiating u with respect to ^, and mul-
tiplying the resulting derivative by -r-.
From the equation (4) we have
d^ _ /dtpV df^ ^d^ dtpd'i/}
dx' ~ \dx I dx dip' dx dx'
^ du
d'lp dtp du d'lp
dx' dx dip dx' '
By the rule just given, we have
T^d'u
dip ' _ d\i dip ^
dx ~ dip' dx'
^du
dtp _ d'u dip
dx ~ dip' dx '
Hence, by substitution and aggregation of like terms,
^ _ d'u fdipy ^u d'lp dip du d'lp
dx' ~ di/Adx I "^ dip' dx' dx + dip dx' ' ^^'
Eepeating the process, we shall find
d'u _ d^/dipy d'li^ d'lp fdipy
dx* ~ dip\dxj + dip' dx' \dxl
^u r^ # /^vn , du d'lp .„.
■*" dip' [sdx' dx^''\dx'i A'^dfd^- y^>
90 TEE DIFFERENTIAL OALOULUa.
Example. Let ua take the case of
M — sin ip,
ip being any function whatever of a;. We may then form the
successive derivatives as follows:
du du dtb ,dib
J- = jT :r- = cos ib^;
dx dtp dx ^ dx
d'u . ,fdip\\ ,d'tp
d*u ,ldip\' „ . .dipd'ip
jdipY „ . .dfd'i/} , , d'f
EXERCISES.
Putting = a function of x, find the first three derivatives
of the following functions of (p with respect to x:
I. u = cos (p. 2. M = <p'
3. M = 0". 4. M = 0".
5. u = log 0. 6. M = e* .
■J. u = sin 30, 8. M = cos 30.
55. Change of the Equicrescent Variable. Let the relation
between y and a; be expressed in the form
X = 4>{y), (1)
and let it be required to find the successive derivatives of y
with respect to x, regarding the latter as the equicrescent.
We may do this by solving (1) with respect to y, and then
differentiating with respect to x in the usual way.
But the method of the last article will enable us to express
the required successive derivatives of y with respect to x in
terms of those of x with respect to y, which we can obtain
SPECIAL OASES OF SU00ES8IYE DERIVATIVES. 91
from (1). By differentiating (1) as often as we please, we
have results of the form
D,^x = ct>"y =^"; [ (2)
B^'x=<t>"'y = x"'.)
etc. etc.
x\ x", x"'f etc., thus representing functions of y,
I'rom § 37, Cor., we have
^ = -J- = - . (3)
dx DyX x'' ^ '
To obtain the second derivative, we hate to differentiate x',
a function of y, with respect to x (§ 54). Thus
3^y 1^ dx' dy_
dx* ~ x"' dy dx'
From (3), ^' = f? = x". ''
dy dy •^
Prom this equation and (3) we have
d^
^ - _ E" - _ .^ ,A^
dx'~ x"~ fdx_y' W
Differentiating again, we find
^ ^ (3x^ dx^__l dx^'\ dy
dx'~\x'* dy x" dy) dx
of^A' <^a; d'z
\dyV ~dvdi?
_ 3x"' - x'x'" _ \dyV dy dy
~ x" ~ ' /dxV
[dyj
The above process may be carried on to any extent. But
many students will appreciate the following more elegant
method of obtaining the required derivatives.
Imagine that we have solved the equation (1) so as to
obtain a result in the form
y = F{x). (5)
92 THE DIFFERENTIAL CALCULUS.
If in this equation we substitute for x its value (1), we shall
have a result in the form
y = F{cPy), (6)
which, of course, will really be an identity.
But we may still differentiate (5) with respect to y, regard-
ing a; as a function of y given by (1), by the method of §§ 40
and 54. Thus we shall have
<?'y _ d'yldxV d'y d'x dx dy d'x
dy' ~ W'Xdy I dx' dy' dy dx dy''
etc. etc. etc. etc.
But from the identity (6) y = y, which is obtained from
(5), we have
^-1- ^-0- ^l-Q- etc
dy - ^' df - "' dy' ~ "' ®*°-
Therefore, substituting for the derivatives of x with respect
toy the expressions x', x", etc., in (4), we have the equations
x'^-1-
"" dx-^'
^ dx'^'^ dx- "'
dx' dx dx '
a;" %! + ^x"x" S + (4a;'a;"' + 3a;'") ^ + x'^% = 0.
dx* dx ' dx' dx
Solving these equations successively, we shall find the values
of -r-, -t4j etc., already obtained.
56. Case of Two Variables Connected by a Third. The
case is stiU to be considered in which the relation between x
8PWIAL OASES OF 8U00E88IVE DERIVATIVES. 93
and y is expressed in the form
y = 4>M; a; = 0,(m). (1)
From these equations it is required to find the successive
deriratiye of y with respect to x.
The first derivative is given by the equation
dy_
dy _ du _ J)„y
dx dx B^x'
du
From the manner in which the second member of this equa-
tion is formed, it is an explicit function of u alone. Hence
(§ 54) we obtain its derivative with respect to x by taking its
derivative with respect to u, and multiplying by -j—. Thus
dx d'y dy d'x
d'y
du du' du du' du
dx''
/dx\' ■ dx
\duj
dx d'y dy d'x
du du' du dAi'
Idx y
\du]
This, again, being a function of u, further derivatives with
respect to x may be obtained by a repetition of the process.
EXERCISES.
Find the second derivative of x with respect to y, and also
of y with respect to x, when the relation of x and y is given
by the following equations:
1. a; = a cos m; y^h sin u.
2. x = a cos 2m; y — I sm u.
3. a; = a cos 2m; y = 5(cos u — sin u).
4. a; = M — e sin m; y = m + e sin m,
5. x^e"; y^ tce?\
94 THE DIFFERENTIAL CALCULUS.
6. Show that if
., d'u 3 sin u
« = e" cos M, then ^-5- = -^, = a-
^ ay e*"(cos u — sm u)
7. Show that the wth derivative of a;" + aa;"~' + 5a;"~' is
n\, n being a positive integer > 1.
8. Show that
X>^'(m') = ^vJ'D^^u + ISuD^uDJ'u + 6(Z>^m)'.
9. Show that if v = m", then
DJ'v - nu^-^DJu + 3n(n - l)u''-^D^uDJ'u
+ n{n - 1) (w - 2)m"-'(Z)^m)'.
10. If ?* = a cos ma; + 1 sin ma;, show that
Z>/m + m'M = 0.
Then, by successively differentiating this result, show that,
whatever the integer n,
i?/ + % + m"i?/M = 0;
11. If M = e* cos a; and v = (f sin x, then
D^'u = -2v and D^'v = 2m.
Also, i)„*t; 4- 4v = 0;
i>^% + 4m = 0.
12. If M = e"'" cos ma; and v = e"* sin ma;,
show that the successive derivatives of m and v may always be
reduced to the form
BJu — Aiu — Bfl; DJv = AtV + BfU, (a)
where A and B are functions of m and w. Also, find the
values of A^, A^ B^ and B^, and show by differentiating (a)
that
A^^^ = A,Ai- B,B^; Bt+^ = B,At + A.B^.
DEVELOPMENTS IN 8EBIE8. 95
CHAPTER VIII.
DEVELOPMENTS IN SERIES.
5 7 . A series is a succession of terms all of whose values
are determined by any one rule.
A series is called
Finite when the number of its terms is limited;
Infinite when the number of its terms has no limit.
The sum of a finite series is the sum of all its terms.
The sum of an infinite series is the limit (if any) which the
sum of its terms approaches as the number of terms added to-
gether is increased without limit.
"When such a limit exists, the series is called convergent.
When it does not exist, the series is called divergent.
To develop a function means to find a series the limit of
whose sum, if convergent, shall be equal to the function.
We may designate a series in the most general way, in the
form
«i + w, + ?*, + .,.+ M„ + u„^i -f . . . ,
the nth. terms being called m„,
58. Convergence and Divergence of Series. ISTo universal
criterion has been found for determining whether any given
series is convergent or divergent. There are, however, a great
number of criteria applicable to a wide range of cases. Of
these we mention the simplest.
I. A series cannot le convergent unless, as n becomes in-'
finite, the ntli term approaches zero as its limit,
For if, in such case, the limit of the terms is a finite
quantity a, then each new term which we add will always
96 THE DIFFEUENTIAL CALCULUS.
change the sum of the series by at least a, and so that sum
cannot approach a limit.
As an example, the sum of the series
1 — 1 + 1 —1 + 1 — 1, etc., ad infinitum,
will continually change from + 1 to 0, and so can approach
no limit, and so is divergent, by definition.
II. A series all of whose terms are positive is divergent
unless nUn = when w = oo .
To prove this, we have first to show that the harmonic
series
i-\-i-{-i + i+ etc., ad infinitum,
is divergent. To do this we divide the terms of the series,
after the first, into groups, the first group being the 2 terms
I + i, the second group the following 4 terms, the third
group the 8 terms next following, and, in general, the nth
group the 2" terms following the last preceding group. We
shall then have an infinite number of groups, each greater
than i.
Now, if, for all the terms of the series after the nth, we
have
nu„ > a {a being any finite quantity),
then w. > — ,
n
andw„ + Mm+i+ ...>«(- + - — rT + - — r~o + • • •)•
' ^ \m m + 1 m-\-2 J
Because the last factor of the second member of this equa-
tion increases to infinity, so does its product by a, which
proves the theorem.
III. If the terms of a series are alternately positive and nega-
tive, continually diminish, and approach zero as a limit,
then the series is convergent.
Let the series be
U^ — U, + U, — U^ + U, — . . . .
Then, by hypothesis,
U^> U,> U^> u^> . . . .
DEVELOPMENTS IN 8EBTE8. 97
Let us put 8n for the sum of the first n terms of the series,
n being any even integer, and 8 for the limit of the sum, if
any there be. Then this limit may be expressed in either of
the forms
8=8„ +(m„ + i-m„ + j) + (m,. + 3-m» + 4) + - • •
and
8= 8n + l— (w„ + 8 — M„ + s) — (Mn + 4 — Mn + s) — • • • •
Since all the differences in the parentheses are positive, by
hypothesis it follows that, how many terms soever we take,
the sum will always be greater than 8„ and less than jS^„+i.
The difEerence of these quantities is «„ + !, which, by hypothe-
sis, approaches zero as a limit. Since the two quantities /S„
and ;S'„+i approach indefinitely near each other from opposite
directions, they must each approach a limit 8 contained be-
tween them.
Graphically the demonstration may be shown to the eye
thus; Let the line 08^ represent the sum 8n, when n = 6,
O Ss Sb Sj»— S Sn S'o Sr
i I I 1 I I
FiQ. 11.
or any other even number; 08^ the sum 8^, etc. Then every
succeeding even sum is greater than that preceding, and
every succeeding odd sum is less than that preceding, while
the two approach each other indefinitely. Hence there must
be some limit 8 which both approach.
An example of such a series is
1 1,11,1 ^
(— 1)"
of which the wth term is — ^r -zr. We shall hereafter see
3m — 1
that the limit of the sum of this series is in: If we divide
the terms into pairs whose sums are negative, the series may
be written >
3 JJ a
3-5 7-9 1113
etc.
98 THE DIFFERENTIAL CALOULUS.
Pairing the terms so that the sum of each pair shall he posi-
tive, the series becomes
_LJ__1_ _l_f
We may show by the preceding demonstration that these
series approach the same limit.
IV. If, after a certain finite number of terms, the ratio of
two consecutive terms of a series is continually less than a cer-
tain quantity a, luhich is itself less than unity, then the series
is convergent.
Let the mth term be that after which the ratio is less than
a. We then have
Mn + i < ocu„;
M» + 8 < «Mn + l < «X;
M» + s < "M„ + a < aX;
Taking the sum of the members of these inequalities, we
have
«„ + l + M„+s + M„ + s+ • • • <(« + «' + «' + • ■ ')u„.
But a + or' + or' + • • • is an infinite geometrical progres-
cc . .
sion whose limit when or < 1 is ^ , .a finite quantity.
1 — a
Hence, putting S for the limit of the sum of the given
series, we have
1 — a
The second member of this inequality being a finite
quantity which S can never reach, 8 must have some limit
less than that quantity.
As an example, let us take the exponential series
DEYEL0PMENT8 IN 8EBIEB. 99
The ratio of the {n + l)st to the wth term is -. This
ratio becomes less than unity when n> x, and it approaches
zero as a limit. Hence the series is convergent for all values
of a;.
CoKOLLAKT. A Series
fl„ + a^x + a^x' + a^x' + . . .
proceeding according to the powers of a variable, x, is conver-
gent when x <1, provided that the coefficients a„ do not in-
crease indefinitely.
Rbmakkb. — (1) Note that, in applying the preceding rule, it does not
suffice to show that the ratio of two consecutive terms is itself always
leas than unity. This is the case in the harmonic series, but the series is
nevertheless divergent. The limit of the ratio must he less than unity.
(3) If the limit of the ratio in question is greater than unity, the series
is of course divergent. Hence the only case in which Rule IV. leaves
a douht is that in which the ratio, heing less than unity, approaches
unity as a limit. But most of the series met with come into this class.
(3) The sum of a limited number of terms of a series gives no certain
indication of its convergence or divergence. If we should compute the
successive terms in the development of e-'™ we should soon find our-
selves dealing with numbers having thirty digits to the left of the deci-
mal-point, and still increasing. But we know that if we should continue
the computation far enough, say to 1000 terms, the positive and negative
terms would so cancel each other that in writing the algebraic sum we
should have 42 zeros to the right of the decimal-point.
On the other hand, if the whole human race, since the beginning of his-
tory, had occupied itself solely in computing the terms of the harmonic
series, the sum it would have obtained up to the present time would have
been less than 44. For 1000 million of people writing'5000 terms a day
for 3 million of days would have written only 10^' terms. It is a theorem
of the harmonic series, which we need not stop to demonstrate, that
T> ^ -..T 1 ^n,^ comm. log 10" 19
But Nap. log 10" = 0.4343 . . ' = .4334^: = *^ ^^'
and yet the limit of the sum of the series is infinite.
100 TBE DIFFERENTIAL CALCULUS.
59. Maclaurin's Theorem. This theorem gives a method
of developing any function of a variable in a series proceed-
ing according to the ascending powers of that variable.
If X represents the variable, and (j) the function, the series
to be investigated may be written in the form
cl>{x) = A,-\-A,x + A,x'-\-A,x'+...; (1)
the series continuing to infinity unless is an entire func-
tion, in which case the two members are identical.
Whether the development (1) is or is not possible depends
upon the form of the function <p. Most functions admit of
being so developed; but special cases may arise in which the
development is not possible. Moreover, the development will
be illusory unless the series (1) is convergent. Commonly this
series will be convergent for values of x below a certain mag-
nitude, often unity, and divergent for values above that mag-
nitude. What we shall now do is to assume the development
possible, and show how the values of the coefiBcients A may be
found.
Let us form the successive derivatives of the equation (1).
We then have
0(a;) = -^o + ^i« + ^,a;' -f etc.;
g= <P'{x) = A, + ZA,x-\.^A,x^+...;
^ = <l>"{x) = 1-2A, + 2-3A,x + 3-iA,x' + . . . ;
-f = c^"'{x) = 1-2-3A, + 2-3-iA,x + .
dx
^ = 0C)(a;) = 1-2-3-4 . . . wJ„ + etc.
By hypothesis these equations are true for all values of x
small enough to render the series convergent. Let us then
put a; = in all of them. We then have
DEVELOPMENTS IN SERIES. 101
^{0)=A,; .•.^„ = 0(O).
<P'(0)=A,; .-.A, = <f>'iO).
0"(O) = l-2^,; ...^, = -l0"(O).
,p"'{0) = 1-2-3A,; .•.A, = :rir5<P"'{0).
1 2^
1
i-a-s"*
0W(O) = n]A„; .-. A^ = ^0<"'(O).
By substituting these values in (1) we shall have the re-
quired development. Noticing that the symbolic forms (p'{0),
<p"{0), etc., mean the values which the successive derivatives
take when we put a; = after differentiation, we see that the
coefficients are obtained by the following rule :
Form the successive derivatives of the given function.
After the derivatives are formed, suppose the variable to be
zero in the original function and in each derivative.
Divide the quantities thus formed, in order, by 1; 1; 1"3;
l'3-3, etc., the divisor of the nth derivative being n\
The quotients will be the coefficients of the powers of the
variable in the development, commencing with the zero power,
or absolute term.
EXAMPLES AND EXERCISES.
I. To develop {a + a;)" = w in powers of x. We have
u = {a-\- xY; . • . ^„ = a".
~ = n{a + xy~^; .•.A,=na''-\
d'u , ,,,,%.., i n{n — l) „ „
-^=n{n-l){a + xY-^; -•-A= \.c^ ^ «""'-
-^= '^^^ - '^) ■ • • ^^ - ^ -^'^'> ^'^ + ^y
102 TEE DrFFEBENTIAL OALGTILXTS.
Thus the development is
(a + xY = a" + «a»-'a; + (j)a"-"a;' + (j)«"~°*' + • • • .
which is the binomial theorem.
2. Develop (a — a;)" in the same way.
3. Develop log (1 + x).
Here we shall have
du
dx~
1
■ 1 + x
(1+^r
1.
>
dx' ~
-(1 + ;
.)--
d'u
dx' ~
1-2(1 +
a:)--
etc.
eto.
Noticing that log 1 = 0, we shall find
log (1 + a;) = a; - ^" + ia;' - K +
4. Develop log (1 — x).
5. Develop cos x and sin x.
The successive derivatives of sin x are cos x, — sin x, — cos x, sin «,
etc. By putting x=Q, tliese become 1, 0, — 1, 0, 1, 0, etc. Tlius we
find
6. Develop e', where e is the Naperian base.
Ans. e« = l+a; + |] + |.J+....
7. Develop e'".
8. Show that
a^ = l + .loga+<i^' + ^^'+....
9. Deduce e"""" = 1 + * + ^-^+
DEVELOPMENTS IN BEBIEB. 103
10. Develop sin (a + x) and cos (a -\-x) and thence, by com-
paring with the results of Ex. 5, prove the formulae for the
sine and cosine of the sum of two arcs. Pind first
x' x'
sin {a-{-x)i= sin a (1 — -j- + . .) + cos a {x — ^+ • ■)•
11. Develop (1 + e")" and show that the result may be re-
duced to the form
lA
M n' -\-n x' w° + 3w° «^
"^3^"^""2^~L2 H 35 3T+* •
12. Develop e* sin x and e°° cos a; and deduce the results
e'sinas^a^ + aly-falj -4|j -8|y -...
e* cos a; = 1 -f- a; — ^jr^ — 4-i-r — 4^7^+,. . .
2! 4! 5!
13. Develop cos" x.
Begin by expressing cos' x in the form J cos 3a! -f 4 cos sb.
14. Develop tan '~%.
This case affords us an example of how the process of de-
velopment may often be greatly abbreviated. It has been
shown that
f?-tan<-*>a; 1 , , , . . , x / x
J = zr-, — 5 = 1 — a' -+- a;* — a; -f etc. (a)
ax 1 -f a;' ^ '
Now assume
tan<-"a; = A -\- A^x + A^x" -\- etc.
This gives
^•*^^""'^ = A, + 2A,x + 3A,x' + etc. {b)
Comparing (a) and (5), we have
^1 = 1; ^s = - h ^. = i; A = - h etc.
and A, = A^ = A^ . . . =0.
The value of A^ is evidently zero. Hence
tan<-«a;=:a;-ia;= + ^a:'-|a;'-f etc. (c)
104 TBE DIFFERENTIAL CALCULUS.
15. Develop sin<~"x.
®"^°® — di = <1 - * )" *'
we may develop the derivative and proceed as in the last ex-
ample. We shall thus find
. ,_ .. X , 1 a;', 1-3 a:', 1-3-5 a;' , ,
sm<% = - + -.3- + ^^.-+^:^^+etc.
60. Ratio of the Circumference of a Circle to its Diameter.
The preceding development of tan^~'^a; affords a method of
computing the number ?r with great ease. The series (c)
could be used for this purpose, but the convergence would be
very slow. Series converging more rapidly may be obtained
by the following device:
Let a, a', a", etc., be several arcs whose sum is 45° = ^tt.
We then have
tan {a + a' + a" + etc.) = 1.
Let t, t', I", etc., be the tangents of the arcs a, a.', a", etc.
If there are but two arcs, a and a', we then have, by the
addition theorem for tangents,
*'^% = \; or t-\-t' = \-tt'.
1-tt'
U there are three arcs, a, a% and a", we replace t' by
f + 1"
in the last expression, and thus get
1 - ft"
t + f + f- tt't" = !-«'- ft" - tt".
We now have to find fractional values of t, f and t" of the
form — , m being an integer, which will satisfy one of these
equations. Unity is chosen as the numerator because the
powers of the fraction are then more easily computed. The
simplest fractions which satisfy the last equation are
*~3' ^ -5' * -8-
DEVELOPMENTS IN 8EBIIS8.
105
We then have, from the development of tan ^~'' t, etc..
"~2 3-3'
5-3'
1
** 5 3-5' ' 5-5'
-T- » = or + or' + a".
4
These series were used by Dase in computing it to 200
decimals.
A combination yet more rapid in ordinary use is found by
determining a and a' by the conditions
1
tan a ■■
5'
AlU — a' = -T- 7t.
4
We then have
tan 2a =
n
f
o
tan 4a; = jj^;
and because a' = 4a — i^r = 4ar — 45°, we have
, , _ tan 4a — 1 _ 1
^^^'^ -tan4a-fl-2r9-
Hence we may compute ar thus :
_3,_J_, J 1_ p
" 5 3-5'"^5-5' 7.5' + ---»
239 3-339= ' 5-339' ' "'
7f = 4a — a'.
Ten or eleven terms of the first series, with four of the
second, will give it to 15 places of decimals.
106
THM DIFFERENTIAL CALCULUS.
61. In developing functions by Maclaurin's theorem we
may often be able to express the derivatives of a certain order
as functions of those of a lower order. The process of find-
ing the higher derivatives may then be abbreviated by retain-
ing the derivatives of lower orders in a symbolic form, so far
as possible.
EXAMPLES.
I. Let us develop
u = log (1 + sin x) = <p{x).
We now have
cos X
sm X
= sec X — tan x;
1 + sin cB cos X
<p"{x) = sec X tan x — sec" a; = — sec a;0'(a:).
Now, in continuing the differentiation, we use the last of
these forms instead of the middle one. Thus
<p"'(x) — — sec a; tan x <p'{x) — sec x<p"{x)
= — sec a; tan x 4>'{x) + sec' x<p'{x)
= - <p\x)4>"{x).
We may now find the successive derivatives symbolically.
Omitting the symbol x after 0, we have
0'" = — 0'0"' — 0'";
(jf = — (p'cf)^^ — 30"0'";
(ff^ = — (p'(f>^ — 4^"0'^ — 3(f>"".
etc.
etc.
Supposing x = 0,
•^(0) = 0;
0'-(O) = - 2;
0'(O) = + 1;
0-(O) = + 5;
0"(O) = - 1;
0-(O) ^ - 16;
0"'(O) = + 1;
etc. etc.
ence
log (1 -j- sin a;) = a; —
a;'
2
x' x' x' x'
"•"6 12+34 45 "^
DEVELOPMENTS IN SERIES. 107
2. To develop u = tan x.
Let us write the equation in the implicit form
u cos X — sin X = 0.
Then, by differentiation and diyision by cos x, we find
B^u = 1 + m';
D^'u = 2uD^u = 2u + 2u';
BJu = 2uDJu + SD^uDJ'u + 6{DJ'uy.
Putting M = 0, we find the even derivatives to vanish and
the odd ones to become 1, 2, 16, etc. Hence
tan x = x -^^x' -{• ^x^ + . . . .
3. To develop m = sec x.
Differentiating the form w cos cb — 1 = 0, we find
DjM cos a; — M sin a; =: 0. (a)
The successive derivatives of this equation may each be
written in the form
M cos X — N sin a; = 0. (5)
For, if we differentiate this equation with respect to x, it
becomes
{D^M - N) cos X- {M+ D^N) sin a; = 0.
Hence the derivative of (5) may be formed by putting
M' = D^M- N; N' = M-\-B^N, (c)
and writing M' and N' instead of M and N in the equation.
In {a) we have
M = D^u; N='U,.
Then, by successive substitution in (c),
Jf = D^'u- u; N' - 2D^u;
M" - D^'u- 3D^u; N" = SD^'u - u;
M'" = DJii- en^'u + u; N"' = 4J)J'u - iD^u;
M'^ = DJu-lODJ'u + 5D^u; W'" - 6DJ'u - lODJ'u + u.
M" = DJu - IbD^'u + 152)^»M - u;
108
THE BIFFEBENTIAL OALOULm.
When a; = 0, we have sin a; = 0, cos x = l, u = 1, and
hence M = M' = . . . = in all the equations. Thus we
find, for X = 0,
DJu = u = 1;
D^'u = 6-1 = 5;
D^'u = 75 - 15 + 1 = 61;
etc. etc. ;
while the odd derivatives all vanish. Hence
^ , 1 . . 5 . , 61 , ,
sec a; = 1 + - k' + j-j a; + gj a; +
63. Taylor's Theorem. Taylor's theorem differs from
Maclaurin's only in the form of stating the problem and ex-
pressing the solution. The prohlem is stated as follows:
Having assigned to a variable z an increment h, it is re-
quired to develop any function of x-\-h in powers of h.
Solution. Let be the function to be developed, and let
nsput
u = 0(a;); |
M' = 0(a;+A). I
Assume
u' = X„ + XJi + X,h' + XJi' + etc
where X„, X^, etc., are functions of x to be determined.
Then, by successive differentiation, we have
du/
dh
d*u'
(1)
(3)
dh'
d'u'
dh'
etc.
: X, + ZXJi + ZXJi' + 4X.7j' + etc.;
: 3X, + 2 • ZX,h + 3 • 4X,/^' + etc. ;
(3)
1-3-3X, + 3-3-4X,A + etc.
etc. etc.
We now modify these equations by the following lemma:
If we have a function of the sum only of several quantities,
the derivatives of that function with respect to those quantities
will be equal to each other.
DEVELOPMENTS IN SEIIIE8. 109
For if in f{x + /*) we assign an increment /lA to x and to
h separately, the results will be /(a; -\-h-{- Ah) and /(a; -f Ah
+ h), which are equal.
It follows that we have
du' _ du'
dh~ dx'
Now these equal derivatives, like u' itself, are functions of
x-\-h alone, so the lemma may be applied to as many suc-
cessive derivatives as we please, giving
d'u' _ d'u\
d¥ ~ dx^ '
ffu' _ d'u'
d¥ ~ dx' '
etc. etc.
Now let the derivatives with respect to x be substituted for
those with respect to h in equations (3), and let us suppose h
to become zero in equations (2) and (3). Then u' and its de-
rivatives will reduce to u and its derivatives, and we shall get
-p. _ du
^■-^'
„ _ J_ ^
'~l-2dx"
Y - 1 ^'^
''~l-2-idx'''
_ 1 d"u
''~ n\ dx"" '
Then, by substitution in (3), we shall have, for the required
development,
, , du h , d'u V , d^u ¥ , ,
" ='* + 5^r+;to^r2+^'FF3+^*«-
This formula is called Taylor's Theorem, after Brook
Taylor, who first discovered it.
110 THE DIFFERENTIAL CALCULUS.
EXAMPLES AND EXERCISES.
1. Develop {x -\- /*)».
We proceed as follows :
u = a;";
du
dx
^^,=n{n- 1) {n - %-';
etc. etc.
By substitution in the general formula we find
{x + hy = a;" + ^a"-' h + ^^" ~ ^^ a;"-" A
1 L' Z
n(n - 1) (w - a) 3,3 ,
2. Develop the exponential function a " + * in powers of h.
Ans. a^il + log a^ + (log a)'^-^ +.••]•
3. sin (x -\- h). 4. cos {x -\- h).
5. sin {x — h). 6. cos (a; — A).
7. log (ce + h). 8. log (a; — A).
, a; + 7i
9. log T. 10. log cos a;.
iJ/ "^ /J
II. cos' {x 4" A). 12. sin' {x — h).
13. tan<-"(a; + A). 14. sin (-'> (a; - 7i).
15. Deduce the general formula
1 6. Prove, by differentiation and appljang the algebraic
theorem that in two equal series the coefficients of like
powers of the variables must be equal, that if we have
log («o + «i» + o,a;' + ...) = J„ + 5,a; +5,a;' + . . . ,
DEVELOPMENTS IN SERIES. Ill
then the coefficients a and b are connected by the relations
\ = log «o;
a A = «>;
%a,\ + afi, = 2a,;
3a^b, + 2a^b, + a,b, - 3a,;
etc. etc. etc.
17. Hence show that is the logarithm of the sum of
an infinite series whose first terms are
3a;' , 13a;' , 73a;* , \
e(l + x + ''
2 ' 6 ' 34 ' • ■ V
63. Identity of Taylor's and Maclaurin's Theorems.
These two theorems, though different in form, are identical
in principle.
To see how Taylor's theorem flows from Maclaurin's, notice
that Ti in the former corresponds to x in the latter. The de-
rivatiyes with respect to x in Taylor's theorem are the same
as the derivatives with respect to h, and if we suppose /» =
after differentiation Taylor's form of development can be de-
rived at once from Maclaurin's.
Conversely, Maclaurin's theorem may be regarded as a
special case of Taylor's theorem, in which we take zero as the
original value of the variable, and thus make the increment
equal to the variable. That is, if we put f{x) in the form
/(O + x),
and then, using x for li, develop in powers of x by Taylor's
theorem, we shall have Maclaurin's theorem.
64. Gases of Failure of Taylor's and Maclaurin's
Theorems. In order that a development in powers of a vari-
ble may have a determinate value it is necessary that none of
the coefficients in the development shall become infinite and
that the developed series shall be convergent.
Por example, cosec x cannot be developed in powers of x,
because when x = the cosecant and all its derivatives be^
come infinite.
112 THE DIFFERENTIAL CALGULUS.
65. Extension of Taylor's Theorem to Functions of Several
Variables. Let us have the function
w =/(», y). (1)
It is required to develop this function when x and y both re-
ceive increments.
Let us first assign to x the increment 7i, and suppose y to
remain constant. We then have, by Taylor's theorem,
. . du h d'u ¥ d'u h' . .
in which u, t— , etc., are all functions of y.
Next, assign to y the increment k. The first member of
(2) will become /(a; -}• 7i, y -\- k). Developing the coefficients
in the second member in powers of k, the result will be:
u will be changed into
du h d'u k' d'u ¥
" + 5^ r + ^' 2 ! + ^= 3 ! + • • • '
-7— = D^^u will be changed into
„ d-D^uh d'Dxu¥
■^"■^ '^'~d^l^~df 2! + • • • ^
d'u
^-i = D^u will be changed into
„, , d-D^u lc , d'DJu lc' ,
■^'='* + ~^%~l+~^'2!+--- 5
etc. etc. etc.
Substituting these changed values of the coefficients in (2)
it will become
., , , , , , , du k , d'u ¥ , d^u ¥ ,
duh d'u li Tc d'u h ¥
'^dxl^ dxdy 1 1 "*" dxdf 1 2 ! + ' ' '
d'u h' d'u ¥ h d'u ¥¥
+ dx' 2! "^ dx'dy^l 1 + ^3;"%" 2! 2!
'^ dx'd\^
DEVELOPMENTS IN SERIES. 113
Thus the function is developed in powers and products of
the increments h and Ic.
The law of the series will be seen most clearly by using the
i?-notation. For each pair of positive and integral values of
m and n we shall have the term
ml n\
If we collect in one line the terms of the development
which are of the same order in It and k, we shall have:
Order of
Terms. , ,
1st. Dji- + DyUj.
2d. i),'M|j + D^D,u\-\ + D,'u^,.
3d. D^'u^, + D^'D.u Ij ^ + D^D^u ||j + D.'u |j;
rth. X>/wi.J+ i),'-ii)„M^_l^ _ +
EXERCISES.
1. Show that in the preceding development the terms of
the rth order may be written in the form
^ I nrD^ru + ^|^^|^r - 1 ^.D^r - IJJ^^ + (^^i •- 'A'^^x*- " ^Dy^V, + . . . L
\t)' (9")' ^^^'' •isnoting the binomial coeflScients as in § 5.
2. Extend the development to the case of three independent
variables, and show that the t??ni? to the second order in-
clusive will be as follows :
114
If
THE DIFFERENTIAL CALCULUS.
u=f{x,y, Z),
then /■{x-\-h,y-\-k,z-\-l)=u
+ D^u-h + DyU-k + D^u-l
+ DjiD^u • hi + DyUD,u ■ kl.
66. Hyperiolic Functions. The sine and cosine of an
imaginary are may be found as follows: In the deyelopments
for sin x and cos x, namely.
sm X ■■
X' , X'
cos a; = 1
3! ^5!
a;' «* _
2! ' 4!
let us put yi for x. (i = V— 1). We thus have
■)■■
sin yi = i\^i + U.J + Ij +
cos yi = 1 + 1; + f I + •
(1)
We conclude:
The cosine of a purely imaginary arc is real and greater
than unity., while its sine is purely imaginary.
We find from (1),
cos yi + i sm yi = 1 — y -\-^. — etc, = e"";
COS yi — i sin 2/i = 1 + y + |-, + etc. = e";
and, by addition and subtraction,
COS yi = \(e~'' -\- e*);
i sin «/i = ■J(e~'' — e");
sin yi = ^i{e^ — e"*).
The cosine of yi is called the hyperbolic cosine of y,
and is written cosh y, the letter h meaning "hyperbolic."
DEVELOPMENTS IN SERIES. 115
The real factor in the sine of yi is called the hyperbolic
sine of y, and is written sinh y.
Thus the hyperbolic sine and cosine of a real quantity are
real functions defined by the equations
sinh?/ = ^{6" — e—");\
coshy = i(e''H-e-''). 5
By analogy^ we introduce the additional function
tanh y ■■
(1)
e^ + e-"
The differentiation of these expressions gives
d sinh y , d cosh y . , ,„,
— ^-■- = cosh 2/; -^-l=smhy; (2)
d tanh y — —
cosh" y'
They also give the relations
cosh" y — sinh" y = 1. (3)
Inverse Hyperbolic Functions. When we form the inverse
function, we may put
u = cosh y.
Then, solving the equation
el -\- e-" = 2 cosh y = 2u,
we find e" = u± V-u
Hence
y = log (u ± Vu' — 1) = cosh ^~*' M. (4)
In the same way, if we put
u = sinh y,
we find
y = log («. ± V'u' + 1) = sinh<-« u. (5)
From the equations (3) and (3) we find, for the derivatives
of the inverse functions:
116
THF, DIFFERENTIAL CALGULUi
When
y — cosh'~'> u, or u = cosh y,
1/1x611
dy _ 1
du Vw> _ 1
When
y — sinh'~^' u, or ^t, = sinh y,
then
dy _ 1
au ^u' + 1
(6)
(7)
Eemaek. The above functions are called hyperbolic be-
cause sinh y and cosh y may be represented by the co-ordinates
of points on an equilateral hyperbola whose semi-axis is unity.
The equation of such an hyperbola is
x'-y' = 1,
which is of the same form as (3).
EXERCISES.
1. By continuing the differentiation begun in (3) prove
the following equations:
Dj.' sinh X = sinh x;
DJ' cosh X = cosh x;
Dx^~^ sinh x = sinh x.
etc. etc.
2. Develop sinh x, as defined in (1), in powers of x byMac-
laurin's theorem.
Ans. sinh a; = y + |-j -f- |j + . . . .
3. Develop sinh (x -\- h) and cosh {x -\- h) by Taylor's
theorem and deduce
sinh (x+h) = sinh a;f 1 + ^, + . . .]+ cosh x\x+^^+ . . . j
= sinh X cosh h + cosh x sinh h;
cosh (x+h) — cosh x cosh h + sinh x sinh h.
MAXIMA AND MINIMA. Ill
CHAPTER IX.
MAXIMA AND MINIMA OF FUNCTIONS OF A
SINGLE VARIABLE.
67. Def. A maximum value of a function is one wMch
is greater than the values immediately preceding and follow-
ing it.
A minimum value is one which is less than the values
immediately preceding and following it.
Eemaek. Since a maximum or minimum value does not
mean the greatest or least possible value, a function may
have several maxima or minima.
68. Peoblem. Having given a function
y = <P{«),
a is required to find those values of x for which y is a maxi-
mum or a minimum.
Let us assign to x the increments -j- h and — h, and develop
in powers of h. We shall then have
, , , , . dy h , d'y ¥ ,
ti -1/ , 7N , dy h , d'y h' , ,
In order that the value oi y = 4>{x) may be a minimum, it
must, however small we suppose h, be less than either y' or
y". That is, the expressions
, dy 7i , d'y V
y -y=-d^i+ii^iii-^^-'
,, , dy h . d'y h" , ,
118 TEE DIFFERENTIAL CALCULUS.
must both be positiye as h approaches zero. But if -^ is
finite, h may always be made so small that the terms in A'
shall be less in absolute magnitude than those in h (§ 14), and
the condition of a minimum cannot be satisfied. We must
therefore have, as the first condition,
|^0'(-) = O. (1)
By solving this equation with respect to x will be found a
value of X called a critical value.
The same reasoning applies to the case of a maximum, so
that the condition (1) is necessary to either a maximum or a
minimum. Supposing it fulfilled, we have
d'y h' d'y ¥ , ,
y -y=i^^v2-di^T^+^^-'
y y- dx'i-^i^ dx'i-%-d^^^^-
Since A' is positive, the algebraic sign of these quantities,
as li approaches zero, will be the same as that of -t4'
When this second derivative is positive for the critical value
of X, y, being less than y' or y", will be a minimum.
When negative, y will be greater than either y' or y", and
so will be a maximum.
We therefore conclude:
Conditions of mininium: -j— = 0; -^^ positive.
Conditions of maximum: -^=0; -^.^ negative.
We have, therefore, the rule:
Eq'uate the first derivative of the function to zero. Tliis
equation will give one or more values of the independent vari-
able, called critical values, and thence corresponding values of
the function.
MAXIMA AND MINIMA.
119
Suhstituie the critical values in the expression for the second
derivative. When the result is positive, the function is a
minimum; when negative, a maximum.
Exceptional Cases. It may happen that the second deriva-
tive is zero for a critical value of x. We shall then have
d'y ¥ , d'y h'
d'y V , d'y h' , ,
: 0. If this condition is fulfilled, y will be a maximum
and there can be neither a maximum nor a minimum unless
dhi
dx''
when the fourth derivative is negative; a minimum when it
is positive.
Continuing the reasoning, we are led to the following ex-
tension of the rule :
Find the ■first derivative in order which does not vanish
for a critical value of the independent variable. If this de-
rivative is of an odd order, there is neither a maximum nor a
minimum; if of an even order, there is a minimum when the
derivative is positive, a maximum when it is negative.
The above reasoning may be illustrated by the graphic rep-
resentation of the function. When the ordinate of the curve
is a maximum or a minimum the tangent will be parallel to the
axis of abscissas, and the angle which it makes with this axis
will change from positive
to negative at a point hav-
ing a maximum ordinate,
and from negative to posi-
tive at a point having a
minimum ordinate.
For example, in the fig- -^^- '^^•
ure a minimum ordinate occurs at the point Q, and maxi-
mum ordinates at P and R.
120 THE DIFFERENTIAL 0ALCULU8.
EXAMPLES AND EXERCISES.
I. Find the maximum and minimum values of the expres-
sion
y = %x'-{- Bz' - 36a; + 15.
By differentiation,
^ == 6a;' + 6a; - 36;
ax
Equating the first derivative to zero, we have the quadratic
equation
a;' + a; — 6 = 0,
of which the roots are x = 2 and a; = — 3.
d'x
The values of -^—^ are + 30 and — 30.
Hence x = 2 gives a minimum value of y = — 29; •
a; = — 3 gives a maximum value ot y = -\- 95.
Find the maximum and minimum values of the following
functions:
2. x' + 3a;' — 242; + 9. 3. x' — 3x + 5.
X x' — X -\-l
log X log X
8. y = af . 9- y = sin 2a; — x.
10. y = {x-\- l){x - 2)'. II. y = {x- a'Xx - h)'.
"•^-(a;+2r '3- y (x - p){x - ^Y
14. y = cos 2a;. 15. y = cos wa;.
16. y = sin 3a;. 17. y = sin wa;.
a; Ans. A maximum when a; = -f-cos x.
' ^ ~ 1 -f- a; tan x' A minimum when a; =— cos a;
MAXIMA AND MINIMA.
121
19. y = sm X cos x. 20. y = sm x cos a;,
sin a; cos x
1 + tan x' '—3 J _j_ ^^Q g."
23. The sum of two adjacent sides of a rectangle is equal
to a fixed line a. Into what parts must a be divided that the
rectangle may be a maximum? Ans. Each part = ^a.
Note that the expression for the area is «(« — x).
24. Into what parts must a number be divided in order
that the product of one part by the square of the other may
be a maximum? Ans. Into parts whose ratio is 1 : 3.
Note that if a be the number, the parts may be called x and a — x.
25. Into what two parts must a number be divided in order
that the product of the with power of one part into the nth.
power of the other may be a maximum?
Ans. Into parts whose ratio is m : n.
26. Show that the quadratic function ax^ -\-'bx-\-c can have
but one critical value, and that it
will depend upon the sign of the
coeflScient a whether that value
is a maximum or a minimum.
27. A line is required to pass
through a fixed point P, whose
co-ordinates are a and b in the
plane of a pair of rectangular
axes OX and OY. What angle
must the line make with the axis
of X, that the area of the triangle XTO maybe a minimum?
Show also that P must bisect the segment XY.
Express the intercepts which the line cuts off from the axes in terms of
a, b and the variable angle a. The half product of these intercepts will
be the area.
We shall thus find
Fio. 13.
9 Area = (d + 5 cot a:)(5 + a tan or) = 3aJ-j- a^ tan a -f-
tan a
122 THE DIFFERENTIAL CALCULUS.
Then, taking tan a — ta& the independent variable, we readily find, for
the critical values of t and a,
J
t = ± - , or assinct=±J cos a.
a
It is then to be shown that both values of t give minima values of the
area ; that the one minimum area is Zab, and the other zero ; that in the
first case the line TX is bisected at P, and in the other case passes
through 0.
28. Show by the preceding figure that whatever be the an-
gle XO Y, the area of the triangle will be a minimum when
the line turning on P is bisected at P.
The student should do this by drawing through P a line making a
small angle with XPT. The increment of the area XOT will then be
the difference of the two small triangles thus formed. Then let the small
angle become infinitesimal, and show that the increment of the area
XOFcan become an infinitesimal of the second order- only when PX=
PT.
29. A carpenter has boards enough for a fence 40 feet in
length, which is to form three sides of an enclosure bounded
on the fourth by a wall already built. What are the sides
and area of the largest enclosure he can build out of his ma-
terial? Ans. 10 X 30 feet = 300 square feet,
30. A square piece of tin is to have a square cut out from
each corner, and the four projecting flaps are to be bent up so
as to form a vessel. What must be the side of the part cut
out that the contents of the vessel may be a maximum?
Ans. One sixth the side of the square.
31. If, in this case, the tin is a rectangle whose sides are
2a and 2i, show that the side of the flap is
i{a + b - Va' -ab + b').
32. What is the form of the rectan-
gle of greatest area which can be drawn
in a semicircle?
Note that if r be the radius of the circle,
and X the altitude of the rectangle, l/r^ — x^
will be half the base of the rectangle.
Fig. 14.
MAXIMA AND MINIMA. 123
69. Oase when the function which is to be a maximum or
minimum is expressed as a function of two or more variables
connected by equations of condition.
The function which is to be a maximum or minimum may
be expressed as a function of two variables, x and y, thus:
u = 4>{x, y). (1)
If X and y are independent of each other, the problem is
different from that now treated.
If between them there exists some relation
A^, y) = 0, (2)
we may, by solving this equation, express one in terms of the
other, say y in terms of x. Then substituting this value of
y in (1), u will be a function of x alone, which we may treat
as before.
It may be, however, that the solution of the equation (2)
will be long or troublesome. We may thfen avoid it by the
method of § 41. Erom (1) we have
du _ ldu\ fdu\dy
dx ~ \dx J \dy Jdx '
and from (3) we have, by the method of § 37,
dy _ BJ
dx ~ Dyf
Substituting this value in the preceding equation, we shall
have the value of -^, which is to be equated to zero. The
equation thus formed, combined with (2), will give the critical
values of both x and y, and hence the maximum or minimum
value of u.
124 THE DIFFERENTIAL CALCULUS.
EXAMPLES AND EXERCISES.
I. To find the form of- that cylinder which has the maxi-
mum volume with a given extent of surface.
The total extent of surface includes the two ends and the convex
cylindrical surface. If r be the radius of the base, and h the altitude,
we shall have :
Area of base, itr'<.
Area of convex surface, %7crh.
Hence total surface = 27r(r' + rh) = const. = a. (a)
Also, voliune = ttj-'/i. (6)
Putting M for the volume, we have, from (J),
du n , , „dh
-T- = 27(rh + nr^^-.
dr dr
I*rom (a) we find
dr~ r '
Whence -^ = Ttrh — 2«r*.
dr
Equating this to zero, we find that the altitude of the cylinder must be
equal to the diameter of its base.
2. Pind the shape of the largest cylindrical tin mug which
can be made with a given weight of tin.
This problem differs from the preceding one in that the top is sup-
posed to be open, so that the total surface is that of the base and con-
vex portion.
Ans. Altitude = radius of bottom.
3. Find the maximum rectangle which can be inscribed in
a given ellipse.
If the equation of the ellipse is bV + aV = a''b\ the sides of the
rectangle are Sa; and 3y. Hence the function to be a maximum is 4a;^,
subject to the condition expressed by the equation of the ellipse. This
condition gives
dp _ h^x
das ~ a^y
MAXIMA AND MINIMA.
125
We shall find the rectangle to be a maximum when its sides are
proportional to the corresponding axes of the ellipse; each side is then
equal to the corresponding axis divided by yS.
4. Find the maximum
rectangle which can be
inscribed in the segment
of a parabola whose semi-
parameter is p, cut ofE by
a double ordinate whose
distance, OX, from the
vertex is a. Show also
that the ratio of its area
to that of the circum-
scribed rectangle is con-
stant and equal to
2: V27.
By taking x and y as in the Fio. 15.
figure, a — X will be the base
of the rectangle, and we shall have 2y for its altitude. Hence its area
will be 32/(a — x), while x and y will be connected by the equation of the
parabola, y'^ = 2px.
5. Find the cone of maximum volume which shall have a
given extent of conical surface.
Ans. Alt. = radius of base X V2.
6. Find the volume of the maximum cylinder which can be
inscribed in a given right cone, and show that the ratio of its
volume to that of the cone is 4 : 9.
7. Find the cylinder of maximum cylindrical surface which
can be inscribed in a right cone.
Ans. Alt. of cylinder = ^ alt. of cone.
8. Find the maximum cone which can be inscribed in a
given sphere.
If we make a central section of the sphere through the vertex of the
cone, the base and slant height of the cone will be the base and equal
126 THE DIFFERENTIAL CALCTTLTTS.
sides of an isosceles triangle inscribed in the circular section. Thus the
equation between the base and altitude of the cone can be obtained.
Ans. Alt. = f radius of sphere.
9. Find the maximum cylinder which can be inscribed in
an ellipsoid of revolution.
Ans. Alt. = —p of axis of revolution.
10. Find the cone of maximum conical surface which can
be inscribed in a given sphere.
r I. Of all cones having the same slant height, which has
the maximum volume ?
12. A boatman 3 miles from the shore wishes, by rowing
to the shore and then walking, to reach in the shortest time
a point on the beach 5 miles from the nearest point of the
shore. If he can pull 4 miles an hour and walk 5 miles an
hour, to what point of the beach should he direct his course?
Ans. 4 miles from the nearest point of the shore.
Express the whole time required in terms of the distance x of his point
of landing from the nearest point of the shore.
13. Find the maximum cone which can be inscribed in a
paraboloid of revolution, the vertex of the cone being at the
centre of the base of the paraboloid.
Ans. Alt. = \ alt. of paraboloid.
14. Find the maximum cylinder which can be described in
a paraboloid of revolution.
15. Find the rectangle of maximum perimeter which can
be inscribed in an ellipse.
16. On the axis of the parabola y' = 2px a point is taken
at distance a from the vertex. Find the abscissa of the near-
est point of the curve.
Begin by expressing the square of the distance from the fixed point to
the variable point (a;, y) on the parabola.
17. Determine the cone of minimum volume which can be
circumscribed around a given sphere.
MAXIMA AND MINIMA, 127
1 8. Determine the cone of minimum conical surface which
can be circumscribed around a given sphere.
19. Find that point on the line joining the centres of two
circles from which the greatest length of the combined cir-
cumferences will be visible.
20. Find that point on the line joining the centres of two
spheres of radii a and h respectively from which the greatest
extent of spherical surface will be visible.
Ans. The point dividing the central line m the ratio «' : h .
21. Show that of all circular sectors described with a given
perimeter, that of maximum area has the arc equal to double
the radius.
22. A ship steaming north 13 knots an hour sights an-
other ship 10 miles ahead, steaming east 9 knots; What will
be the least distance between the ships if each keeps on her
course, and at what time will it occur?
Ans. Time, 32 min.; distance, 6 miles.
23. What sector must be taken from a given circle that it
may form the curved surface of a cone of maximum volume?
Ans. Vf of the circle.
24. A Norman window, consisting of a rectangle sur-
mounted by a semicircle, is to admit the maximum amount
of light with a given perimeter. Show that the base of the
rectangle must be double its altitude.
128 THE DIFFERENTIAL CALCULUS.
CHAPTER X.
INDETERMINATE FORMS.
70. Let us consider the fraction
0(^) = ^| (1)
For any value we may assign to x there will be a definite
value of (p{x) found by dividing the numerator of the frac-
tion by the denominator.
To this statement there is one exception, the case of a; = 3.
Assigning this value to x, we have
0(3) = f.
Now, the quotient of two zeros is essentially indeterminate.
For the quotient of any two quantities is that quantity
which, multiplied by the divisor, will produce the dividend.
But any quantity whetever when multiplied by will pro-
duce 0. Hence, when divisor and dividend are both zero,
any quantity whatever may be their quotient.
But when we consider the terms of the fraction, not as ab-
solute zeros, but as quantities approaching zero as a limit,
then their quotient may approach a definite limit. We then
regard this limit as the value of the fraction corresponding
to zero values of its terms.
As another example, consider the quantity
We may compute the value of this expression for any value
of X except 2. When x = % the terms will both become in-
finite. Since if any quantity whatever be added to an infinite
INDETERMINATE P0BM8. 129
the sum will be infinite, it follows that any quantity what-
ever may be the difference of two infinites.
There are several other indeterminate forms. The follow-
ing are the principal ones which take an algebraic form:
^; -; OXoo; oo - oo ; 0°; oo"; 1".
11. Evaluation of the Form -g. In many cases the inde-
terminate character of an expression may be removed by
algebraic transformation. For example, dividing both terms
of the fraction (1) by a; — 3, it becomes x -\- 3, a determinate
quantity even for x = 3. Again, the expression (3) can be
reduced to the form — — -r, which becomes i when x — 2.
x-\-%
The general method of dealing with the first form is as
follows: Let the given fraction be
0(£)
f(x)'
and let it be supposed that both terms of this fraction vanish
when X — a, so that we have
0(a) = and ^(a) - 0. (3)
Put h=x — a, and develop the terms in powers of A by
Taylor's theorem. We shall then have
4>(x) = cp{a + h) = 0(«) + A0'(fl) + ^ct>"{a) + . . . ;
fix) = i>{a + h) = f{a) + hf'{a) + ^f"{a) + . . . ;
whence, for the value of the fraction (comp. Eq. (3)),
(4)
Now, when h approaches zero as a limit, the value of this
fraction approaches
0>)
f\a)
130 THE DIFFERENTIAL CALCULUS.
as a limit, which is therefore the required limit of the frac-
tion when both its members approach the limit zero.
It may happen that <p'{d) and ^p'{a) both vanish. In this
case the required limit of the fraction in (4) is seen to be
,p"{a)'
In general: The required limit is the ratio of the first pair
^/derivatives of like order which do not loth vanish.
If the first derivative which vanishes is not of the same
order in the two terms, — for example, if, of the two quantities
0'(a) and ^'{a), one vanishes and the other does not, — ^then
the limit of the fraction will be zero or infinity according as the
vanishing derivative is that of the numerator or denominator.
Eemaek. It often happens that the terms of the fraction
can be developed in the form (4) without forming the succes-
sive derivatives. It will then be simpler to use this develop-
ment instead of forming the derivatives.
EXAMPLES AND KXERCISES.
a;' -a'
for X = a.*
X — a
<f>{x) -x' - a'; <p'(x) = 2x; - • . (p'{a) = 2a;
ip{x) =x — a; f{x) = 1; .-. f'{a) = 1.
x' — a'
. ' . lim. (x = o) = 2a,
X — a^ '
a result readily obtained by reducing the fraction to its lowest
terms.
log a;
x-\
e -e
X
for x = \. Ans. 1.
-X
— for x = 0. Ans. 2.
* Using strictly the notation of limits, we should define the quantity
sought as the limit of the fraction when x approaches the limit a. But
no confusion need arise from regarding the limit of the fraction as its
value for x = a, asis customary.
INDETERMINATE FORMS. 131
a; — sin a ,
4- i — -- for (x = 0). Ans. ^.
Here the successive derivatives of the terms are:
^'(x) = 1 — cos x; tj)"{x) = sin x; (p"'{x) = cos x.
ib'lx) = 3x'; i/j"{x) = 6x; f"'{x) = 6.
The third derivatives are the first ones which do not vanish
for X = 0.
Q^ 7)^ ff
5- for a; = 0. Ans. loga— logJ=logj^.
X
^ tan a; — sin a; .
6. r for x~V). Ans, 3.
a; — sm a;
7. ; '■ for a; = 0. Ans. -,.
1 — cos n% n
8. L- for a; = 1. Ans. a log a.
a; — 1 ^
a" Ja:
9- T- for a; = 1. ^«s. a log a — J log 5.
sin a; — sin a , ,
lo. for %■= a, Ans. cos a.
X — a
tan « — tan a , , see' a
II. — r^- r~ for y = a. Ans. ^r—. — .
cos y — cos a ^ 3 sm a
log (1 + a;) + log (1 - a;) ^
cosi-sef -^ f°^ ^ = «- ^4«^-+l-
13. H(« + ^)-I°g(«-^) fo, ^ = 0. Ans. I
a; a
sin 2a; + 3 sin' a; — 2 sin a; .
14- 5 for x — 0. Ans. 4.
cos X — cos X
e°—e-''—2x, - .
15. : for X = 0. Ans. 1.
X — sm X
, c" + sin « — 1 . „ . „
1 6. ■ ' .^ f — r— for « = 0. Ans. 2.
log (1 + y)
1 — sin a; — cos a; + log (i + a;) . ,
17. ^_iJa; *°'' (^ = ®)- ^"«- 0-
132 THE DIFFERENTIAL CALCULUS.
12. Forms — and X oo . These forms may be reduced
00
to the preceding one by a simple transformation. Any frac-
tion -=,- may be written in the form ., ' .., . If N and D both
become infinite, 1 -=- Z> and 1 -i- N will both become infini-
tesimal, and thus the indeterminate form of the fraction will
bet
Again, if of two factors A and B, A becomes infinitesimal
while B becomes infinite, we write the product in the form
jj, and then it is a fraction of the first form.
But this transformation cannot always be successfully ap-
plied unless the term which becomes infinite does so through
haying a denominator which vanishes. For example, let it
be required to find the limit of
a;"'(log xY
for a; = 0. Here a;" approaches zero, while log x, and there-
fore (log a;)", becomes infinite for x = 0. Hence the denomi-
nator of the transformed fraction will be ^ (putting for
brevity I = log x). The successive derivatives of this quantity
with respect to x are
The successive derivatives of the numerator are
mx''~'^; m{m — l)x'"~^; etc.
The limiting values of the given quantity a;"?" thus become
mx'^l"*^ m{m — l)x'"
n
'' (J- ^ !L±i\ '
etc..
which remain indeterminate in form how far soever we may
carry them.
INBETEBMINATE F0BM8. 133
In such cases the required limit of the fraction can be
found only by some device for which no general rule can be
laid down. In the example just given the device consists in
replacing a; by a new variable y, determined by the equation
log a; = - y.
We then have a; = e ~ ".
Since for a;i02/=oo,we now have to find the limit of
(-y)" ^ / y.n^
for y = OS.
By taking the successive derivatives of the two terms of
w"
the fraction ~, we have the successive forms
ny^^_ n{n — l) .y"~^. n{n — 1) {n — 2)y"-^ _
me*"" ' wiV""" ' jw^e*"" ' ^ °'
Whatever the value of n, we must ultimately reach an ex-
ponent in the numerator which shall be zero or negative, and
then the numerator will become « ! if w is a positive integer,
and will vanish for y i oo , if «. is not a positive integer. But
the denominator will remain infinite. We therefore con-
clude:
lim. [a:"'(log a;)"] (a; i 0) = 0,
whatever be m and oi, so long as m is positive.
From this the student should show, by putting z = x~^ and
m — 1, that the fraction
z
(log z)"
becomes infinite with z, how great soever the exponent n, and
therefore that any infinite numier is an infinity of higher
order than any power of its logarithm.
73. Form oo — cxi . In this case we have an expression of
the form
F{x) = u — V,
134 TEE DIFFERENTIAL CALCULUS.
in which both u and v become infinite for some value of x.
Placing it in the form
we see that F{x) will become infinite with u unless the fraction
V
— approaches unity as its limit. When this is the case the
expression takes the form oo x of the preceding article.
74. Form 1". To investigate this form let us find the
limit of the expression
when n becomes infinite. Taking the logarithm, we have
log u = hn log ^1 + -j
Making n infinite, we have
lim. log u = h;
or, because the limit of log u is the logarithm of lim. u,
log lim. u = Ti.
I IN*"
Hence lim. 1 1 + - 1 (w = oo ) = e*.
In order that this result may be finite, h itself must not be
infinite. We therefore reach the general conclusion:
Theobbm. In order that an expression of the form
(1 + «)'
may have a finite limit when a becomes infinitesimal and z
infinite, the product ax must not become infinite.
Cor. If the product ax approaches zero as a limit, the
given expression will approach the limit unity.
inbetehminate forms. 135
75. Forms 0" and oo°. Let an expression taking either
of these forms as a limit be represented by u'^=F. The
problem is to find the limiting value of the expression when
(p approaches zero and u either approaches zero or becomes
infinite.
From the identity w = e'"*"
we derive ^ = m* = e* '■« ".
We infer that the limit of F will depend upon that of log m.
If lim. <f) log M is + °° > then lim. F= co.
If lim. log M is — 00 , then lim. F=0.
If lim. <p log M is 0, then lim. F= 1.
If lim. log u is finite, then lim. F is finite.
Hence the rule: To find the limit of m* when i and
w — or 00 , put I £ lim. ^ log u. Then
lim. M* — e'.
EXAMPLES AND EXERCISES.
1. Find lim. af for x = 0.
Here ar" = e"" ">«'".
Since z log a; has zero as its limit when x = 0, the required
limit is e° or 1.
2. lim. a;"^ for x = 0. Ans. F=l.
3. lim. x" for a; i 00.
1
4. a;'-"" for x = l.
n
5. a;!-" for a; = 1.
h
6. (1 — a;)^ for x = 0.
7. = 7^-j — r for a; = 0.
log (1 + x)
„ log sin 2a; . . > 1 o
8. -~ — -. for a; = 0. Ans. log 2.
log sin X
e- + log(l-^) zJ: fo, ^^0. ^«s. i.
• a; — tan a;
Ans.
F=l.
Ans.
1
e ■
Ans.
«-".
Ans.
e-\
Ans.
2.
136 THB DIFFERENTIAL OALGTJLTJB.
' ° ' for a; = 00.
m"
. Tt It
II. z tan a; — — sec a; for x = rr-.
. a ,
12. y sin — for y z= <x>.
13. x\a^ — l) for a; = 00 .
for a; = 0.
1
for x — 0.
Ans.
1.
Ans.
- 1.
Ans.
a.
Ans.
logo.
Ans,
1.
Ans.
et.
Ans.
e-K
Ans.
2
n
/tan a;\
/tan a;\x»
16. (cos a;)x' for a; = 0.
17. (1 - y) tan |«/ for y = \.
1
18. ^^_2SfV for a; = 0. ^ws. 1.
19. a; — a;' log f 1 -| j for a; = 00. ^ws. \.
gr g - a;
20. = T- r- for a; = 0. ^ms. 2.
log (1 + X)
2
' T^ ' J f or a; = 0. Ans. a^a^.
n
22. (-J — ■ — ^— ' ■ — ^1 for a; = 0. Ans. a^a,...an.
23, Show that, how great soever the exponent n,
7i r- iz 00 when a; i 00 .
(log a;)"
PLANE CUBVBa. 137
CHAPTER XI.
OF PLANE CURVES.
"76. Forms of the Equations of Curves. As we have here-
tofore considered curve lines, they have been defined by an
equation between the co-ordinates of each point of the curve,
and therefore of one of the forms
y=f{x); x=f{yy, (1)
and F{x, y) = 0.
The distinguishing feature of the equation is that when we
assign a value at pleasure to one of the co-ordinates x or y,
one or more corresponding values of the other co-ordinate are
determined by the equation.
But the relation between x and y may be equally well
defined by expressing each of them as a function of an
auxiliary variable, which is then the independent variable.
Calling this auxiliary variable u, the equations of a curve will
be of the form
y = <p,{n). f ^^>
Assigning values at pleasure to u, we shall have correspond-
ing values of x and y determining each point of the curve.
An advantage of this method of representation is that for
each value of u we have one definite point of the curve, or
several definite points when the equations give several values
of the co-ordinates for each value of u; and we thus have a
relation between a point and the algebraic quantity u.
It is also to be remarked that by eliminating u from the
equations (3) we shall get a single equation between x and y
which will be the equation of the curve in one of the forms
(1).
138
THE DIPPERENTIAL CALCULUS.
Example 1. Let us put
a, 5 = the co-ordinates of any fixed point 5 of a straight line;
a = the angle which
the line makes with
the axis of z;
p = the distance of
any point P of the
line from the point
{a, I).
Then we readily see
from the figure that
the co-ordinates x and
y ot P are given by the equations
x = a -\- p cos a;
y = b -{- p sin a
■X
X .
- ^
p/
y
a
b/
v
y^
y
y 6
o
A
1
X
y
y
y
Fio. 16.
;}
(3)
^'1
(4)
which are equations of the straight line in the independent
form.
Here p is the auxiliary yariable, called u in Eq. 3. By
eliminating this quantity we shall have
X sin ot — y cos a = a sin a — i cos a,
which is the equation of the line in one of its usual forms.
Example 3. The equation of a circle may be expressed in
the form
a; = a + c cos «;
y = b -\- csinu;
u being the independent
variable.
By writing (4) in the
form
X — a=: c cos u,
y — h = c sin u,
and eliminating u by
taking the gum of the
squares of the two equa-
tions, we have fio. it.
PLANE CURVES. 139
the equation of a circle of radius c.
Notice the beautiful relation between (3) and (4). They
are the same in form: if in (4) we write p for c and a for
u, they will be the same equations. Then, by supposing p
constant and a variable, we are carried round the point (a, t)
at a constant distance p, that is, around a circle. By suppos-
ing p variable and a constant we are carried through (a, V)
in a constant direction, that is, along a straight line.
7 1 . Infinitesimal Elements of Curves. Let P and P' be
two points on a curve, P being supposed
fixed, and P' variable. We may then sup-
pose P' to approach P as its limit, and in-
quire into the limits of any magnitudes
associated with the curve.
We may also measure the length of an
arc of the curve from an initial point G to
a terminal point P. Then, supposing G fixed and P variable,
PP' may be taken as an increment of the arc.
If we put
s = arc GP,
we shall have
As = arc PP'.
Axiom. The ratio of an infinitesimal element of a curve
to the straight line joining its extremities approaches unity as
its limit.
We call this proposition an axiom because a reaUy rigorous
demonstration does not seem possible. Its truth will appear
by considering that if the curve has no sharp turns, which
we presuppose, then it can change its direction only by an in-
finitesimal quantity in any infinitesimal portion of its length.
Now, a line which has the same direction throughout its length
is a straight line.
140
TEE DIFFERENTIAL GALCULU8.
78. Theokem I. If a straight line touch a curve at the
point P, a point P' on the
curve at an infinitesimal
distance will, in general,
he distant from the tangent
iy an infinitesimal of the
second order.
Let y = f {x) be the _o .
equation of the curve.
Leb us transform the equation to a new system of co-ordi-
nates, x' and y', so taken that the axis of JT shall be parallel
FiQ. 19.
to the tangent at P
dv'
This will make j-, = 0. Let x' and y'
be the co-ordinates of P, and {x' -f h, y") the co-ordinates
of a point P' near P.
Developing by Taylor's theorem, we have
y
-' dx' ^ dx'
1-2
+ ..
Now, y" — y'is the distance P'Q of the point P' from the
tangent at P. Since
dl
dx'
0, when h becomes infinitesimal
dy h'
the term of highest order in this distance is t^^ ^Hy, a quan-
tity of the second order.
Eemabe. In the special case when
d^y'
dx" '
: 0, the distance
in question may be a quantity of the third or of some higher
order, according to the order of the first differential coeffi-
cient which does not vanish.
CoEOLLAET. The cosine of an infinitesimal arc differs
from unity hy an infinitesimal of the second order.
For if we draw a unit circle with its tangent at the initial
point, the cosine of an arc will differ from unity by the dis-
tance from the end of the arc to the tangent line. When the
arc is infinitesimal, the coroUary follows from the theorem.
PLANE 0UBYE8. 141
Thboeem II. The area included between an infinitesimal
arc and its chord is not greater than an infinitesimal of the
third order.
From Th. I. we may readily see that the maximum distance
between the chord and its arc is a quantity of the second
order. The area is less than the product of this distance by
the length of the chord, which product is an infinitesimal of
at least the third order.
TO. Expressions for Elements of Curves. Def. An
element of a geometric magnitude is an infinitesimal por-
tion of that magnitude.
The word implies that we conceive the magnitude to be
made up of infinitesimal parts.
Element of an Arc. Let us put
s = the length of any arc of a curve;
ds = an element of this arc.
If P and P' be two points of a curve, we shall have
(chord PPy = ^x' + Ay\
When PP' becomes infinitesimal, as
the ratio of ds to PP' becomes unity
(§ 77), and we have y^ Aa;
ds'' = dx^ + dy''; fio- 20-
ds = Vdx' + df = yi + (^'dx.
Case of Polar Co-ordinates. To express the element of a
curve referred to polar co-ordinates, difEerentiate the equa-
tions
x = r cos 6; y — r sin 0.
Thus dx = cos ddr — r sin 6d0;
dy = sin ddr -\- r cos Odd;
which gives ds'' = dr^ -{- r^dfP
and ds = |/r'+Q'^i9.
142
THE DIFFERENTIAL CALCULUS.
80. Equations of certain Noteworthy Curves. The Cycloid,
The cycloid is a curve described by a point on tbe circumfer-
ence of a circle rolling on a straight line. A point on the
circumference of a carriage-wheel, as the carriage moyes,
describes a series of cycloids, one for each revolution of the
virheel.
To find the equation of the cycloid, let P be the generating
point. Let us take the line on which the circle rolls as the
axis of X, and let us place the origin at the point where P
is in contact with the line OX.
FlO. 21.
Also put
a = the radius of the circle ;
u = the angle through which the circle has rolled, expressed
in terms of unit-radius.
Then, when the circle has rolled through any distance OR,
this distance will be equal to the length of the arc PR of the
circle between P and the point of contact R, that is, to au.
We thus have, for the co-ordinates of the centre, C, at the
circle,
X = au;
y = a;
and for the co-ordinates of the point P on the cycloid.
a; = «M — a sin M = a{u — sin u);
y = a — a cos M = «(! — cos u)
;}
(1)
which are the equations of the cycloid with w as an independ-
ent variable.
PLANE CURVES. 143
To eliminate u, find its value from the second equation,
u = cos(-« [l - ^\
This gives sin u=: Vl — cos' u = —.
° a
Then, by substituting in the first equation
.a — :
x = a cos<-» ^ - V2ay-y', (3)
Cb
which is the equation of the cycloid in the usual form.
81. The Lemniscate is the locus of a point, the product of
whose distances from two fixed points (called foci) is equal
to the square of half the distance between the foci.
Let us take the line joining the foci as the axis of X, and
the middle point of the segment between the foci as the
origin. Let us also put c = half the distance between the
foci.
Fio. 23.
Then the distances of any point {x, y) of the curve from
the foci are
V{x - cf + y^ and V{x + c)' + y\
Equating the product of these distances to c', squaring and
reducing, we find
{x^ + yy^2c^{x^-f), (3)
which is the equation of the lemniscate.
144
THE DIFFERENTIAL CALOULUS.
Transforming to polar co-ordinates by the substitutions
a; = r cos d,
y ^^r sin B,
we find, for the polar equation of the lemniscate,
r' = 3c' cos %e. (4)
Putting y = 0, we find, for the point in which the curve
cuts the line joining the foci,
a; = ± ^c = a.
The line a is the semi-axis of the lemniscate. Substitut-
ing it instead of c, the rectangular and polar equations of the
curve will become
(x' + fy = a'ix'-y');l
r" = a' cos 2(9. ) ^ '
83. The Archimedean Spiral, This curve is generated
by the uniform motion of a point along a line revolving uni-
formly about a fixed point.
To find its polar equation, let us take the fixed point as the
pole, and the position of the revolving line when the generat-
ing point leaves the pole
as the axis of reference.
Let us also put
a = the distance by
which the generating
point moves along the
radius vector while the
latter is turning through
the unit radius.
Then, when the ra-
dius vector has turned
through the angle 6, the
point will have moved
from the pole through the distance aff.
r = a0
as the polar equation of the Archimedean spiral.
Fig. 23.
Hence we shall have
PLANE CURVES.
145
If we increase by an entire revolution (2;r), the corre-
sponding increment of r will be 27ra, a constant. Hence:
The Archimedean spiral cuts any fixed position of the ra-
dius vector in an indefinite series of equidistant points.
83. The Logarithmic Spiral. This is a spiral in which
the logarithm of the radius vector is proportional to the angle
through which the radius vector has moved from an initial
position. Hence, if we put 6^
for the initial angle, we have
log r = i{e- ex
I being a constant. Hence
r = e"'-"'
Putting, for brevity,
a — e °,
the equation of the logarith-
mic spiral becomes
r =
a and I being constants.
Fio. 24.
EXERCISES.
1. Show (1) that the maximum ordinate of the lemniscate
is ic, and (2) that the circle whose diameter is the line join-
ing the foci cuts the lemniscate at the points whose ordinates
are a maximum.
2. Find the following expression for the square of the dis-
tance of a point of a cycloid from the starting point ( 0, Pig. -
21):
r' = 2ay + 2uax — a'u'.
3. A wheel makes one revolution a second around a fixed
axis, and an insect on one of the spokes crawls from the cen-
tre toward the circumference at the rate of one inch a second.
Find the equation of the spiral along which he is carried.
146 TEM DIFFEBENTIAL CALCULUS.
4. If, in that logarithmic spiral for which a = 1 and 1 = 1,
r = e*,
the radius vector turns through an arc equal to log 2, its
length will be doubled.
5. If, in any logarithmic spiral, one radius vector bisects
the angle between two others, show that it is a mean propor-
tional between them.
6. Show that the pair of equations
z = au*,
y = lu,
represent a parabola whose parameter is — .
7. If, in the equation of the Archimedean spiral, 6 and
therefore r take all negative values, show that we shall
have another Archimedean spiral intersecting the spiral given
by positive values of ^ in a series of points lying on a line at
right angles to the initial position of the revolving line.
This should be done in two ways. Firstly, by drawing the continua-
tion of the spiral when, by a negative rotation of the revolving line, the
generating point passes through the pole. It will then be seen that the
combination of the two spirals is symmetrical with respect to the vertical
axis. Secondly, by expressing the rectangular co-ordinates of a point of
the spiral in terms of 8 we have
x — aB cosB,
y =: 06 sin 8.
Changing the sign of 8 in this equation will change the sign of x and
leave y unchanged.
8. Show that if we draw two lines through the centre of a
lemniscate making angles of 45° with the axes, no point of
the curve will be contained between these lines and the axis
of Y.
TANGENTS AND NORMALS.
147
CHAPTER XII.
TANGENTS AND NORMALS.
84. A tangent to a curve is a straight line through two
coincident points of the curve.
Fio. 25.
A normal is a straight line through a point of the curve
perpendicular to the tangent at that point.
The subtangent is the projection, TQ, upon the axis of
X, of that segment TP of the tangent contained between
the point of contact and the axis of X,
The subnormal is the corresponding projection, QN, of
the segment PNoi the normal.
Notice that a tangent and a normal are lines of indefinite
length, while the subtangent and subnormal are segments of
the axis of abscissas. Hence the former are determined by
their equations, which will be of the first degree in x and y,
while the latter are determined by algebraic expressions for
their length.
But the segments TP and PN are sometimes taken as
lengths of the tangent and normal respectively, when we con-
sider these lines as segments.
148 THE DIFFERENTIAL OALOULUS.
85. General Equation for a Tangent. The general prob-
lem of tangents to a curve may be stated thns:
To find the condition which the parameters of a straight
line must satisfy in order that the line may be tangent to a
given curve.
But it is commonly considered in tbe more restricted form:
To find the equation of a tangent to a curve at a given point on
the curve.
Let {x^, «/,) be the given point on the curve. By Analytic
Geometry the equation of any straight line through this point
may be expressed in the form
y-y,=m{x- x,); (5)
m being the tangent of the angle which the line makes with
the axis of X. But we have shown (§ 20) that
m = -f-',
aa;,
this differential coefficient being formed by differentiating the
equation of the curve. Hence
y-&. = ^(^-^.) (6)
is the equation of the tangent to any curve at a point (a;,, y^j
on the curve.
Equation of the Normal. The normal at the point (a;,, y,)
passes through this point, and is perpendicular to the tangent.
If m' be its slope, the condition that it shall be perpendicular
to the tangent is (An. Geom.)
m'=- — =- —
m dy^
dx^
Hence the equation of the normal at the point (a;,, y,) is
dx,
{y - y.) ^00,-x. (7)
TANGENTS AND NORMALS.
149
In these equations of the tangent and normal it is necessary
to distinguish between the cases in which the symbols x and
y represent the co-ordinates of points on the tangent or nor-
mal line, and those where they represent the given point of
the curve. Where both enter into the same equation, one set,
that pertaining to the curve, must be marked by suflBxes or
accents.
86. Suhtangent and Subnormal. To find the length of
the subtangent and subnormal, we have to find the abscissa
a;, of the point T in which the tangent cuts the axis of abscis-
sas. We then have, by definition.
Fig. 26.
Subtangent = x^ — z„
The value of «„ is found by putting y = and a; = a;, in
the equation of the tangent. Thus, (6) gives
y.
dx.
K - «,)•
Hence, for the length of the subtangent TQ,
Subtangent = a;, — a;„ = -1^.
dx^
We find in the same way from (7), for QN,
Subnormal = — y,-^.
^'dx.
(8)
(9)
150 THE DIFWEBENTLAL CALCULUS.
87. Modified Forms of the Equation. In the preceding
discussion it is assumed that the equation of the curre is given
in the form
y =/(«=)•
But, firstly, it may
be given in the form
n^, y)
= 0.
We shall then have
(§37)
dF
dx.
dx^
' dF'
dy.
Substituting this value in the
equations (6)
and 1
Tangent
dF,
yd = a^i?=. -
-x);
-^ , dF, . dF, .
Normal: —(y-y^) = —{x-x,).
(10)
Secondly, if the curve is defined by two equations of the
form
X = 0.(m),
X = <p,{u), I
y = 0,(m), )
(11)
dy^
, dy. du
we bave -^ = -rr-,
du
in which there is no need of suflBxes to x and y in the second
member, because this member is a function of u, which does
not contain x or y.
By substitution in (6) and (7), we find
Eq. of tangent: {y - y,)-£- = (« - a;,)^.
Eq. of normal: (y - y.)^ = {x, - x)^.
(12)
TANGENTS AND NORMALS. 151
By substituting in these equations for x^y^, t— and -^
their values in terms of u, the parameters of the lines will be
functions of u. Then, for each value we assign to u, (11)
will give the co-ordinates of a point on the curve, and (13)
will determine the tangent and normal at that point.
88. Tangents and Normals to the Conic Sections. Writing
the equation of the ellipse in the form
oy + 5V = a'l\ (a)
we readily find, by differentiation,
dy _ Vx
dx ~ a'y'
Applying the suflSx to x and y, to show that they represent
co-ordinates of points on the ellipse, substituting in (6) and
(7), and noting that k, and y, satisfy (a), we readily find:
For the tangent: ^ -f ^ = 1.
For the normal: —x « = o' — J'.
Taking the equation of the hyperbola,
- aY + JV = a'd%
we find, in the same way.
For the tangent: ^ _ M = i.
a
a' J'
For the normal: —x-\ — w = a'-}-J*.
Taking the equation of the parabola,
y' = 2pa;,
we find, by a similar process.
For the tangent: y,y = p{x -\- a;,).
For the normal: y -~ yi= — -(», — x).
152 THE DIFFEBBNTIAL CALCULUS.
89. Problem. To find the length of the perpendicular
dropped from tUe origin upon a tangent or normal.
It is shown in Analytic Geometry that if the equation of a
straight line be reduced to the form
Ax+By+ 0=0,
the perpendicular upon the line from the origin is
G
p =
VA' + 5"
It must be noted that in the above form the symbol rep-
resents the sum of all the terms of the equation of the line
which do not contain either x or y.
If we have the equation of the line in the form
y-y^ = m{x- a;,),
we write it mx — y — ma;, -f y^ = 0,
and then we have
A = m;
B=-l;
G=y, -ma;,.
Thus, the expression for the perpendicular is
y, - mx,
Vm' + i
Substituting for m the values already found for the tan-
gent and normal respectively, we find.
For the perpendicular on the tangent:
</^ + (|)"
For tJie perpendicular on the normal:
x+v^^
„ _ ' 'dx, _ x,dx, + y ,dy.
^^^^ir
TAJraENTS AND NORMALS.
153
Fig. 37.
90. Tangent and Normal in Polar Oo-ordinates.
Pboblem. To find the
angle which the tangent at
any point makes with the
radius vector of that point.
Let PP' be a small arc
of a curve referred to polar
co-ordinates;
KP, a small part of the
radius vector of the point
P (the pole being too far
to the left to be shown in
the figure);
K'P', the same for the point P'.
KSR, a parallel to the axis of reference. Drop PQlK'P'.
Let SPThe the tangent at P. We also put
y = angle KPS ■vrhich the tangent makes with the radius
vector.
Then let P' approach P as its limit. Then
QP' = dr; PQ = rdd;
PQ rdd
^^^y^qP^lTr--
We also have
1 1
(1)
cos y =
•"'+'-■ "'Vi^+^yf
dr
-d'e''
sin y = cos y tan y —
/1''+©T
(2)
Cor. The angle RSP which the tangent makes with the
^xis of reference is y -\- 6.
164 TBE DIFFERENTIAL CALCULUS.
91. Perpendicular from the Pole upon the Tangent and
Normal. When y is the angle between the tangent and the
radins vector, we readily find, by geometrical construction,
that the perpendicular from the pole upon the tangent and
normal are, respectively,
p =-r sin y and p =■ r cos y.
Substituting for sin y and cos y the values already found,
we have.
For the perpendicular on tangent :
p.
For the perpendicular on normal :
(3)
_ r dr
93. Problem. To find the equation of the tangent and
normal at a given point of a curve whose equation is expressed
in polar co-ordinates.
It is shown in Analytic Geometry that if we put
p = the perpendicular dropped from the origin upon a line;
a=the angle which this perpendicular makes with the
axis of X;
the equation of the line may be written
X cos a -\- y sin a — p = 0. (1)
Now, as just shown, the tangent makes the angle y -\- &
with the axis of X, and the perpendicular dropped upon it
makes an angle 90° less than this. Hence we have
a = y + - 90°;
cos a = sin (y -\- 6) = sin / cos ^ + cos y sin 6;
BID. a = — COB {y -{- d) — — cos y cos ff -^ sin y sin d.
TANGENTS AND NOBMALS. 155
By substitution in (1), the equation of the tangent becomes
a;(sin ^^ cos 6* + cos y sin 6)
— y{coB Y cos 6 — sin y sin ^) — ^ = 0,
Substituting for cos y, sin y and^ the values already found,
this equation of the tangent reduces to
[r cos 6 -{--,-„ sin d\x-\-\r sin ^ — -^a cos ^j «/ — r' = 0, (2)
r and 6* being the co-ordinates of the point of tangency.
In the case of the normal the perpendicular upon it is
parallel to the tangent. Therefore, to find the equation of
the normal, we must put in (1)
a=y-{-e.
Substituting this yalue of a, and proceeding as in the Case
of the tangent, we find, for the normal,
(-72 cos 5 — rsin (9ja;-f frcos 6 -\- --rj,%va. d\y — rjo = 0. (3)
Generally these equations will be more conyenient in use if
we divide them throughout by r. Thus we have:
Equation of the tangent :
(''"^ ^ + r ri'''' ^J'' + r"" ^ ~ fS""^ e^y-r = 0. (4)
Equation of the normal :
/I dr
\rdd''°'
0-sm0)x+ (i g sin + cos e)y-% = 0. (5)
In using these equations it must be noticed that the co-
efiicients of x and y are functions of r and 6, the polar co-
ordinates of the point of tangency. When r, and -^^ are
da
given, this point and the tangent through it are completely
determined.
156 THE DIFFERENTIAL CALCULUS.
EXERCISES.
I. Show that in the case of the Archimedean spiral the
general expressions for the perpendiculars from the pole upon
the tangent and normal, respectiyely, are
a<9' , ad
V(i + ff') — ^ ~ V(i + e")'
Thence define at what point of the spiral the radius rector
makes angles of 45° with the tangent and normal. Find also
what limit the perpendicular upon the normal approaches
as the folds of the spiral are continued out to infinity.
Show also from § 92 that the tangent is perpendicular to
the line of reference at every point for which
r sin ^ — a cos 61 = 0,
. and "hence that, as the folds of the spiral are traced out to
infinity, the ordinates of the points of contact of such a tan-
gent approach ± a as their limit.
2. Show by Eq. 13 that in the case of the logarithmic
spiral the angle which the radius vector makes with the tan-
gent is a constant, given by the equation
tan y = J-
3. Show from Eq. 13 that if a curve passes through the
pole, the tangent at that point coincides with the radius
vector, unless -^ = at this point. Thence show that in the
lemniscate the tangents at the origin each cut the axes at
angles of 45°.
4. Show that the double area of the triangle formed by a
tangent to an ellipse and its axes is . Then show that the
a^.y.
area is a maximum when — ' = ± f-'.
a
Show also that the area of the triangle formed by a nor-
mal and the axes is a maximum for the same point.
ASYMPTOTES AND SINGULAB POINTS. 157
CHAPTER XIII.
OF ASYMPTOTES, SINGULAR POINTS AND
CURVE-TRACING.
93. Asymptotes. An asymptote of a curve is the limit
which the tangent approaches when the point of contact re-
cedes to infinity.
In order that a curve may have a real asymptote, it must
extend to infinity, and the perpendicular from the origin upon
the tangent must then approach a finite limit.
Por the first condition it suffices to show that to an infi-
nite value of one co-ordinate corresponds a real value, finite
or infinite, of the other.
For the second condition it suffices to show that the expres-
sion for the perpendicular upon the tangent (§§ 89, 91) ap-
proaches a finite limit when one co-ordinate of the point of
contact becomes infinite. If, as will generally be most con-
venient, the equation of the curve is written in the form
F{x, y) = 0, (1)
the value (1) of the perpendicular, omitting suffixes, may be
reduced to
dF OF
{Gfy+(f)T
\dy
If this expression approaches a real finite limit for an
infinite value of x or y, the curve has an asymptote.
If the curve is referred to polar co-ordinates, we use the
expression (3), § 91, for j». If this approaches a real finite
limit for an infinite value of r, the curve has an asymptote.
158
THE DIFFERENTIAL CALCULUS.
The existence of the asymptote being thus established, its
equation may generally be found from the form (10), § 87,
which we may write thus:
dF , dF
^ + -j-y '■
dx,
dF dF
(3)
by supposing a;, or y, to become infinite.
dF dF
Commonly the coefficients -r- and t— will themselves be-
•' dx^ dy,
come infinite with the co-ordinates. We must then divide
the whole equation by such powers of a;, and y, that none of
the terms shall become infinite.
Fio. 26.
Then the equation be-
94. Examples of Asymptotes.
1. F{x) = x'+ y'- 3axy = O.(fl)
The curve represented by this
equation is called the Folium of
Descartes. The equation (3) gives
in this case, applying suffixes,
= < + y* - 2«a;,y, = ax,y,.
To make the coefficients of x and
y finite for a;, = oo , divide by a;,y,.
comes
\y, xj ^ W. yj^ ^ '
Let us now find from (a) the limit of y, for a;, = oo . We
have
X X
The second member of this equation will approach zero as
a limit, unless y, is an infinite of as high an order as x,',
which is impossible, because then the first member of the
equation containing y,' would be an infinite of higher order
ASYMPTOTES AND SINQULAB POINTS.
159
than the second member, which is absurd. Hence, passing
to the limit.
lim.^|j (a;, = «) = -!.
Then, by substitution in (J), we find, for the asymptote,
» + y + « = 0,
%. Take next the equation
F{x, y) = x' — %x^y — ax^ — a'y = 0. (a)
With this equation (3) becomes
(3a;.' - 4:X,y, - 2ax,)x - (2x^' + a')y
= 3x' - 6x'y^ - %ax* — a'y,. (J)
Pio. 89.
We notice that the terms of highest order in the second
member are three times those of highest order in (a). From
(a) we have
x'-2x,'y =fla;,' + ay.
Substituting in the second member of (J), and dividing by
a;,", (i) becomes
(^-^l:-l>-K?V=''+^- <*')
Solving (a) for y, we find
y, ^ g.' - ax,
x, 2a;,' + a"
an expression which approaches the limit -J when x, = co.
Thus, passing to the limit, {b') gives, for the equation of the
asymptote,
x — %y=a.
160 TOE DIFFERENTIAL CALCULUS.
3. The Witch of Agnesi. This curve is named after the
Italian lady who first investigated its
properties. Its equation is
x'y -\- a'y — a' = 0. (a)
The equation of the tangent is
2x^y^x + {x: + a')y = Zx.'y, + a'y, = 3a' - 2a'y,. (&)
By solving (a) for x and y respectively we see that a:, may
become infinite, but that y, is always positive and less than a.
Hence, to make the coefficient of y in {b) finite for a;, = oo ,
we must divide by x^, which reduces the equation of the
asymptote to
Hence the axis of x is itself an asymptote.
95. Points of Inflection. A point of inflection is a point
where the tangent inter-
sects the curve at the
point of tangency.
It is evident from the
figure that in passing
along the curve, and con-
sidering the slope of the ^'°- *'•
tangent at each point, the point of inflection is one at which
this slope is a maximum or a minimum. Because we have
slope = -rr-,
^ ax
the conditions that the slope shall be a maximum or minimum
are
and -=-^ different from zero. If the first condition is fulfilled,
but if -=^, is also zero, we must proceed, as in problems of maxi-
ASYMPTOTES AND SINGULAR POINTS. 161
ma atid minima, to find the first derivative in order which
does not vanish. If the order of this derivative is even, there
IS no point of inflection for -j^ = Oj if odd, there is one.
As an example, let it be required to find the points of in-
flection of the curve
xy' — a'{a — x).
Eeducing the equation to the form
wefind % = -^y'
The condition that this expression shall vanish is
^xy^ = a',
which, compared with the equation of the curve, gives, for the
co-ordinates of the point of inflection,
3 a
X = -ra; y = ± —T=-.
4: ' " 4/3
EXERCISKS.
Find the points of inflection of the following curves ;
1. xy = a" log — . Ans.
° a
ix=: a{l — cos m);
\y = a
( .K = ae'.
(y = %ae~\
y = a{nu + sin m).
_ (w + l)a.
Ans. '
2/=«(cosC-«(-^)
+
Vr?^^l\
162
THE DIFFERENTIAL CALCULUS.
..-/
Fio. 82.
96. Singular Points of Curves. If we conceive an infini-
tesimal circle to be drawn round any
point of a curve as a centre, then, in
general, the curve will cut the circle in
two opposite points only, which will
be 180° apart.
But special points may sometimes be
found on a curve where the infinitesimal circle will be cut in
some other way than this: perhaps in more or less than two
points; perhaps in points not 180° apart. These are called
singular points.
The principal singular points are the following:
BouMe-poinis j at which a .
curve intersects itself. Here the
curve cuts the infinitesimal circle
in four points (Fig. 33).
Ousps; where two branches of
a curve terminate by touching
each other (Pig. 34). Here the
infinitesimal circle is cut in two coincident points.
Stopping Points; where a curve suddenly
ends. Here the infinitesimal circle is cut in
Fia. 35.
only a single point. ^_^
Isolated Points; from which no curve proceeds, so K_^
that the infinitesimal circle is not cut at all, fio. 36.
Salient Points; from which proceed two branches making
with each other an angle which is neither zero nor 180°.
Here the infinitesimal circle is cut in two points which are
neither apposite nor coincident.
There may also be multiple-points, through which the curve
passes any number of times. A double-point is a special kind
of multiple-point.
A multiple-point through which the curve passes three
times is called a triple-point.
Fis. 33.
Fig. 34.
ASTMPTOms AND SINGULAB POINTS.
163
X
>-
/
Xo I Q^
;
o
/ !
/ 1
s.
97. Condition of Singular Points. Let (a;„, yj be any
point on a curve, and let it be required to investigate the
question whether this point is a singular one. We first trans-
form the equation of the
curve to one in polar co-
ordinates having the point
(a;„, y^ as the pole. To do
this we put, in the equation
of the curve,
x = x„-\- pcosB;) ^jj
y = y. + P sin ft )
The resulting equation
between p and d will be the Fia. 37.
equation of the curve referred to (a;,, y„) as the pole. More-
over, if we assign to /o a fixed value, the corresponding value
of 6 derived from the equation will be the angle 6 showing
the direction QP from Q to the point P, where the circle of
radius p cuts the curve. The limit which 6 approaches as p
becomes infinitesimal will determine the points of intersection
of the 'infinitesimal circle with the curve.
If, now, the given equation of the curve is
F{x, y) = 0,
then, by the substitution (1), the polar equation vdll be
F(x, + p cos e,y,+p sin 6) = 0. (2)
Now, let us develop this expression in powers of p by Mac-
laurin's theorem. Since p enters into (2) only through x and
y in (1), we have
dF dFdx , dFdy JtF , . JLF _,
dp dx dp dy dp dx dy
(because -^- = cos 6 and -r- = sin ^j.
Then
164 TEE DIFFERENTIAL CALCULUS.
d'F dF' JO'Fdx , d'F dy\
■i-r = —r- = cos » T-r ^- + 3— j- -4-
dp dp \dx dp dxdy dpi
4. sin ei— ^^ d'Fdy\
[dxdy dp dy' dp)
= cos" 0-j-,- + 2 sm ^ cos 6-r^r- + sm ^-r-,- = -^ •
Noting that when p = then a; = x„, we see that the de-
velopment by Maclaurin's theorem will be
F{x, y) = Fix,, y,) + p[cos ^g + sin ^g)
+ etc. = 0.
dF dW
Here -5— means the value of -5— when x„ is put for a;, etc.
Because (a;„, y„) is by hypothesis a point on the curve, we
have F{x,, y,) = 0, and the only terms of the second member
are those in p, p', etc. Thus the polar equation (3) of the
curve may be written
F/p + F/'p' + F/"p' + etc. = 0, )
or F/ + F,"p + i?;"'p' + etc. = 0. f ^"^
To find the points in which the curve cuts a circle of radius
p, we have to determine ^ as a function of p from this equa-
tion. When p is an infinitesimal, all the terms after the first
will be infinitesimals. Hence, at the limit, where p becomes
infinitesimal B must satisfy the equation
dF
which gives tan 6 = — — ~.
^.
This is the known equation for the slope of the tangent at
(a;„, yj, and gives only the evident result that in general the
ASYMPTOTES AND SINGULAR POINTS. 165
curve cuts the infinitesimal circle along the line tangent to
the curve at Q.
But, if possible, let the point (a;„y„) be so taken that
J^ = 0; §^ = 0. (4)
dx, ' dy„
Then we shall have FJ = 0, and the equation (3) of the
curve will reduce to
F:'P + F:"p' + etc. = 0,
or F," + F,"'p + etc. = 0.
Again, letting p become infinitesimal, we shall have at the
limit
Dividing throughout by cos' 6, we shall have a quadratic
equation in tan 6, which will have two roots. Since each
value of tan 6 gives a pair of opposite points in which the
curve may cut the infinitesimal circle, and since (5) depends
on (4), we conclude:
The necessary condition of a double-2)oint is that the three
equations
shall be satisfied by a single jmir of values of x and y.
If the two values of tan derived from FJ' = are equal,
we shall have either a cusp, or a point in which two branches
of the curve touch each other. If the roots are imaginary,
the singular point will be an isolated point.
98. Examples of Double-points. A curve whose equation
contains no terms of less than the second degree in x and y
has a singular point at the origin. Eor example, if the equa-
tion be of the form
F(x, y) = Px' + Qxy + By' = 0,
then this expression and its derivatives with respect to x and
y will vanish for x = Q and y = 0.
166
TEE DIFFERENTIAL OALCULm.
Let us now investigate the double-points of the curve
(y' - a'Y - 3aV - 2ax' = 0. (1)
We have
dF
dx
dF
dy
= -6(a'ar4-aa;')
Qax{a -f x);
= ^y{y^ - a') = 4«/(y + «) (y - a).
(3)
The first of these derivatives vanishes for a; = or — a;
The second of these derivatives vanishes f or y = 0, — a or + a.
Of these values the original equation is satisfied by the fol-
lowing pairs:
a;„= 0; 0; -a;)
y,= -a; + a; 0;) ^^^
which are therefore the co-ordinates of singular points.
FlO. 38.
Differentiating again, we have
Forming the equation F" = 0, it gives
ASYMPTOTES AND SINGULAR POINTS. 167
(12^' - 4a'') tan' (? = 6a' + nax.
Substituting the pairs of co-ordinates (3), we find:
At the point (0, — a), ian.e= ±^V2;
At the point (0, + a), tan l9 = ± ^ V3;
At the point (— a, 0), ia.n 6 — ± l/f.
The Talues of tan 6 being all real and unequal, all of these
points are double-points. The curve is shown in the figure.
Eemabk. In the preceding theory of singular points it is
assumed that the expression (2), § 97, can be developed in
powers of p. If the function F is such that this development
is impossible for certain values of x^ and y„, this impossibility
may indicate a singular point at (a:,, y^.
99. Curve-tracing. We have given rough figures of va-
rious curves in the preceding theory, and it is desirable that
the student should know how to trace curves when their
equations are given. The most elementary method is that of
solving the equation for one co-ordinate, and then substitut-
ing various assumed values of the other co-ordinate in the
solution, thus fixing various points of the curve. But un-
less the solution can be found by an equation of the first or
second degree, this method will be tedious or impracticable.
It may, however, commonly be simplified.
1. If the equation has no constant term, we may sometimes
find the intersections of the curve with a number of lines
through the origin. To do this we put
y = mx
in the equation, and then solve for x. The resulting values
of a; as a function of m are the abscissas of the points in which
the curve cuts the line
y — mx = 0.
Then, by putting
m= ±1, ±2, etc.; in = ± i, ± \, etc.,
we find as many points of intersection as we please.
168 THE DIFFERENTIAL GALCULU8.
To make this method practicable, the equations which we
have to solve should not be of a degree higher than the second.
If the curve has a double-point, it may be convenient to
take this point as the origin.
2. If the equation is symmetrical in z and y ot x and — y,
the curve will be symmetrical with respect to one of the lines
X — y ^=Q and a; + y = 0.
The equation may then be simplified by referring it to
new axes making an angle of 45° with the original ones.
The equations for transforming to such axes are
x={x' -\- y') sin 45°;
y — {x' — y') sin 45°.
Application to the Folium of Descartes. If, in the equa-
tion of this curve,
x' -\-y' = 3axy,
we put y = mx, we shall find
Sam 3am'
^ — 1 _L ^y> y — ■
l + w" ^ l+m='
We also find, from the equation of the curve and the pre-
ceding expressions for x and y in terms of m,
dy _x' — ay _2m — m*
dx ax
-y'~
1 - 2m''
Then, for
m =
1,
3
y =
3
2"'
^y -^ 1
dz
m —
3,
2
y =
4
3^;
dy 4
dz 5"
m =
3
2'
36
" = 35^'
y =
54
dy 33
dz ~ 92'
m= —
■2,
6
x = ^a;
y = -
12
dy 20
dx ~ 17"
etc.
etc.
etc
,
etc.
Thus we have, not only the points of the curve, but the
tangents of the angle of direction of the curve at each point,
which will assist us in tracing it.
THEORY OF ENVELOPES. 169
CHAPTER XIV.
THEORY OF ENVELOPES.
100. The equation of a curve generally contains one or
more constants, sometimes called parameters. For example,
the equation of a circle,
{x - ay + {y- by = r\
contains three parameters, a, h and r.
As another example, we know that the equation of a
straight line contains two independent parameters.
Conceive now that the equation of any line, straight or
curve, (which we shall call "the line" simply,) to be written
in the implicit form
^{«, y, «) = 0, (1)
a being a parameter. By assigning to a the several values
a, a', a", etc., we shall have an equal number of lines whose
equations will be
0(«, y, oi) = 0; (t>{x, ij, a') = 0; ^{x, y, a") = 0; etc.
The collection of lines that can thus be formed by assign-
ing all values to a parameter is called a family of lines.
Any two lines of the family, e.g., those which have a and
a' as parameters, will in general have one or more points of
intersection, determined by solving the corresponding equa-
tions for X and y. The co-ordinates, x and y, of the point of
intersection will then come out as functions of a and a'.
Suppose the two parameters to approach infinitesimally near
each other. The point of intersection will then approach a
certain limit, which we investigate as follows;
170 THB DIFFERENTIAL CALCULUS.
Let us put
a' ■= a-\- Aa,
The equations of the lines will then be
<p{x, y, a) =0 and ^(x, y, a-\- /la) = 0.
If we develop the left-hand member of the second equation
in powers ot Aahj Taylor's theorem, it will become
<t>{x, y, a) + -^Aa + ^ 3-3 + etc. = 0.
Subtracting the first equation, dividing the remainder by
Aa, and passing to the limit, we find
d(l>{x, y, a)
da
0.
Hence the limit toward which the point {x, y) of intersec-
tion of two lines of a family approaches as the difference of
the parameters becomes infinitesimal is found by determining
z and y from the equations
<Pix,y,a)^0 and M^^) = 0. (2)
The values of x and y thus determined will, in general, be
functions of a; 'that is, we shall have
a= =/.(«); 2/ =/.(«); (3)
which will give the values of the co-ordinates x and y of the
limiting pojnt of intersection for each value of a,
Jfow, suppose a to vary. Then x and y jn (3) will also
vary, and will determine a curve as the locus of x and y.
Such a curve is called the envelope of the family of
lines, (j){x, y, a) = 0,
In (3) the equations of the curve are in the form of (2),
§ 76, a being the auxiliary variable. By eliminating a either
from (2) or (3), we have an equation between x and y which
wiU be the equation of the curve in the usual form.
THEORY OF ENVELOPES. Ill
lOl. Theorem. The envelope and all the lines of the
family which generate it are tangent to each other.
Oeometrically the truth of this will be seen by drawing a
series of lines varying their position according to any con-
tinuous law, as in the first example of the following sec-
tion. Taking three consecutive lines and numbering them
(1), (3) and (3), it will be seen that as (1) and (3) approach
(2) their points of intersection with (3) approach infinitely
near each other. Since these infinitely near points of inter-
section also belong to the envelope, the line (3) passes through
two infinitely near points of the envelope and is therefore a
tangent to the envelope.
Analytic Proof. The equation of the envelope is found by
eliminating a from the equations (3), and we may conceive
this elimination to be efEected by fin.ding the value of a from
the second of these equations (3), and substituting it in the
first equation. That is, the equation
<j){x, y, a) = (4)
represents any line of the original family when we regard a
as a constant; and it represents the envelope when we regard
a as a function of x and y, satisfying the equation
d<p{x y,a)^^^
da ^ '
Let the value of a derived from this last equation be
a = F{x, y). (6)
Now, to find the slope of the tangent to the original line of
the family at the point {x, y), we difEerentiate (4), regardiug
« as a constant. Thus we have
^ I ^^ 1^ = or ^ = - ^
dx dy dx dx Dy<p' ^ '
If the original line is a straight one, this equation will give
its slope.
To find the slope of the tangent to the envelope at the saffie
172
THE DIFFERENTIAL CALCULUS.
point, we diflEerentiate this same equation, regarding a as hav-
ing the value (6), Thus we have
d<p d<p dy
Ax dy dx
d(p
d^fda da dy\ _
+ d^w + ^ ^y ~ ^^'
But, because -p- = 0, this equation will also give the value
(7) for the slope; whence the curves have the same tangent at
the point {x, y), and so are tangent to each other at this point.
103. We shall now illustrate this theory by some examples.
1. To find the envelope of a straight line which moves so that
the area of the triangle which it forms with the axes of co-
ordinates is a constant.
Fig. 39.
Since the area of the triangle is half the product of the
intercepts of the axes cut off by the line, this product is also
constant.
Calling a and 5 the intercepts, the equation of the line may
be written in the form
0(a;, 2/, a) = I + 1 - 1 = 0.
(1)
THEORY OF ENVELOPES. 173
Here we haye two varying parameters, a and h, while, to
have an envelope, the change of the parameters must depend
on a single varying quantity. But the condition that the
product of the intercepts shall be constant enables us to elimi-
nate one of theparameters, say i. We have, by this condition,
h = -, (2)
, dh c
whence t— = ..
da a
Now differentiating the equation (1) with respect to a, re-
garding 5 as a function of a, we have
d<j) _ X y dh _cy — Vx _y x
c a'
da~ a' b' da ~ a'b' ~ - -' ~ ^' ^^^
We have now to eliminate a from the equations (1) and (3),
using (3) to eliminate b from (1). The easiest way to effect
this elimination is as follows:
From (3) we have
I ex
a'y = cx; a = W--. (4)
Multiplying (1) by a, and substituting for b its value from
(3), we have
xA — a.
c
Substituting from (4), this equation becomes
V y
and thus the equation of the envelope becomes
xy = Ic
which is that of an hyperbola referred to its asymptotes.
This result coincides with one already found in Analytic
Geometry, that tangents to an hyperbola cut ofp from the
asymptotes intercepts whose product is a constant.
174
THE niFFEBENTIAL CALCULVS.
2. To find the envelope of the line for which the sum of the
intercepts cut off from the co-ordinate axes is a constant.
Fio. 40.
Let c be the constant sum of the intercepts. Then, if a be
the one intercept, the other will be c — a. Thus the equa-
tion of the line is
X
a
-^ = 1,
c — a
in ■which a is the varying parameter.
Clearing of fractions, we may write the equation
<p{x, y, a) = ex -\- a{y — x — c) -\- a' = 0,
d^
da
whence
-~ = y -x-c-^-ia-O.
From the last equation we haye
a-\{x-y-^ c);
this value of a being substituted in the other gives
cx — l{x-y-\- cf = 0,
or (x — y)' — 2c(x + y) + c' = 0.
THEOBT OF ENVELOPES. 175
This equation, being of the second degree in the co-ordi-
nates, is a conic section.
The terms of the second degree forming a perfect square,
it is a parabola.
The equation of the axis of the parabola is
a; — y = 0.
To find the two points in which the parabola cuts the axis
of X we put y = 0, and find the correBponding values of x.
The resulting equation is
x' — 2cx + c' = 0.
This is an equation with two equal roots, x =^ c, showing
that the parabola touches the axis of X at the point (c, 0).
It is shown in the same way that the axis of Y is tangent to
the parabola.
It may also be shown that the directrix and axis of the
parabola each pass through the origin, and that the parame-
ter is V2c.
3. If the difference of the intercepts cut off by a line from
the axes is constant, it may be shown by a similar process
that the envelope is still a parabola. This is left as an exer-
cise for the student, who should be able to demonstrate the
following results :
(a) When the sum of the intercepts is a positive constant,
the parabola is in the first quadrant ; when a negative con-
stant, the parabola is in the third quadrant.
{/3) When the difference, a — l, of the intercepts is a posi-
tive constant, the parabola is in the fourth quadrant; when a
negative constant, in the second.
(/) The co-ordinate axes touch the parabola at the ends of
the parameter.
In each case the parabola touches each co-ordinate axis at
a point determined by the value of the corresponding inter-
cept when the other intercept vanishes, and each directrix
intersects the origin at an angle of 45° with the axis.
176 THE DIFFERENTIAL OALCULUS.
4. Next take the case in wMch the sum of one intercept
and a certain fraction or multiple of the other is a constant.
Let m be the fraction or multiplier. We then have
I -{-ma = c = a. constant.
The equation of the line then becomes
a c — ma
Proceeding as before, we find the equation of the envelope
to be
{mx — yf — 2c{mx + y) -|- c' = 0,
which is still the equation of a parabola.
5. To find the envelope of a line which cuts off intercepts
subject to the condition
^+5.=!, («)
m and n being constants.
We may simplify the work by substituting for the varying
intercepts a and b the single variable parameter a determined
by either of the equations
m n
sm a = — ; cos a = 7-.
a
The equation of the varying line will then oecome
Mx, «) = — sin or + — cos a = 1. (1)
By differentiating with respect to a, we have
T— = — cos a — — sm a = 0. (2)
da m n
We may now eliminate a by simply taking the sum of the
squares of these equations, which gives
the equation of an ellipse whose semi-axes are m and n.
TEEORT OF ENVELOPES. 177
6. To find the envelope of a circle of constant radius whose
centre moves on a fixed circle.
For convenience let us take the centre of the fixed circle as
the origin, and put:
a, b, = the co-ordinates of the centre of the moving circle;
c = its radius;
d = the radius of the fixed circle.
The equation of the moving circle now becomes
(z - a)' + {y- ly - c' = 0. (1)
By differentiation with respect to a,
x-a+{y-i)^ = ().
The condition that (a, h) lies on the fixed circle gives
a' + J' = <f% (2)
, dl a
whence t- = — r-
da
Then, by substituting this value,
ay — lx = 0. (3)
We have now to eliminate a and h from (1), (2) and (3).
Firstly, from (1) and (2), we find
a;' + 2^' - %ax - %by = c' - d\ {!')
From (2) and (8) we find the following expressions for a
and 5:
xd , yd
a = , ==r; J = ■—===.
Va' + f Vx' + y'
By substitution in (1'), and putting for brevity
r' = x' + y\
we find r' ± 2rd + d' = c'.
Hence r' = x' + y' = (c ± df,
the equations of two circles around the origin as a centre,
with radii c -\- d and c — d.
178 THE DIFFEBENTIAL CALCULUS.
7. Find the envelope of a family of ellipses referred to their
centre and axes, the product of whose semi-axes is equal to
a certain constant, c".
-4ms. The equilateral hyperbola xy = ^c'.
8. To find the envelope of a family of straight lines, such
that the product of their distances from two fixed points is a
constant.
Let (ffl, 0) and (— a, 0) be taken as the two fixed points,
and let c^ be the constant. Also, let
X cos a -\- y sin a — p ■=<) (1)
be the equation of any one of the lines in the normal form,
p and a being the varying parameters.
The distances of the line from the points (a, 0) and (—a,
0) are respectively
— p -\- a cos a and — p — a cos a.
Hence we have the condition
jo' — a' cos' a = c'. (2)
DifEerentiating (1), regarding p asa function of a, we have
From (2) we obtain
— X sm a 4- y cos a — ,
" da
dp __ a' sin a cos a
da~ P '
thus have the three equations
a; cos a + y sin « = p.
(a)
a' sin a cos a
xsm a — y cos a = ,
(S)
p' = c' + a' cos' a
— (^-\-a* — a* sin' a.
(c)
from which to eliminate p and a.
-r «. y
P '
sm a
•
P
X
cos a
c' + a'
P '
y
sm a
THEORY OF ENVSLOPMS. 179
To effect the elimination of a and p we find the values of
K and y from (a) and {b) by taking
(a) X cos a + (J) x sin a and (a) X sin a — (S) X cos a.
We thus find, by the aid of (c),
px = p' cos a-]- a' sin" or cos a;
, , , ,,cos a
x= {c^ -\- a')
Hence
c -j- a'
K we multiply the first of these equations by x and the
second by y.and add, then we have
a;' , y' _ * cos a + y sin or _
cM^"? "*" ? ~ ;? ~
Hence the equation of the envelope is
This represents an ellipse whose foci are the two fixed
points.
This interpretation, however, presupposes that the product
c' of the distances of the line from the two points is positive;
that is, that the points are on the same side of the enveloping
line. If the product is negative, the equation of the envelope
will be
y' _ 1
C c-
which is the equation of an hyperbola.
These results give the theorem of Analytic Geometry that
the product of the distances of a tangent from the foci of a
conic is constant.
180 THE DIFFERENTIAL CALCULUS.
CHAPTER XV.
OF CURVATURE; EVOLUTES AND INVOLUTES.
103. Position; Direction; Curvature. The position of
any point P on a curve is fixed by the values of the co-ordi-
nates, X and y, of P. This is shown in Analytic Geometry.
If we have given, not only x and y, but the value of -^ for
the point P, then such value of the derivative indicates the
direction of the curve at the point P, this direction being the
same as that of the tangent at P,
The curve may also have a greater or less degree of curvo'
ture at P. The curvature is indicated by a change in the di-
rection of the tangent, that is, in the value of —-, when we
pass to an adjacent point P', But such change in the value
of ■— when we vary x is expressed by the value of the second
derivative -r^. If this quantity is positive, the angle which
the tangent makes with the axis of ^is increasing with x at
the point P, and the curve, viewed from below, is convex.
d'ti
If -^ is negative, the tangent is diminishing, and the
curve, seen from below, is concave.
To sum up: If we take a value of the abscissa x, then the
corresponding value of
y gives the position of a point P of the carve;
-^ gives the direction of the curve at P;
-^ depends upon the curvature of the curve at P.
CURVATURE; E VOLUTES AND INVOLUTES. 181
104. Contacts of Different Orders. Let two different
curves be given by their respective equations:
y =f{^) aiid y = (j>{x).
If for a certain value of x, whicli value call a;„, the two
values of y are equal, the two curves have the corresponding
point in common; that is, they meet at the point (a;„, y).
If the values of ~ are also equal at this point, it shows
that the curves have the same direction at the point of meet-
ing. They are then said to touch each other.
d'v
If the values of -=-^ are also equal at this point, the two
curves have also the same curvature at this point.
To show the result of these several equalities, let us give
the abscissa a;„ (which we still take the same for both curves),
an increment h, and develop the two values of y in powers of
h by Taylor's theorem. To distinguish the values of y, -r-,
etc., which belong to the two curves, we assign to one the
suflSx 0, and to the other the suffix 1. Then, for the one
curve,
and, for the other,
'■-.+(l),^+(S^),^.+--+(g)|+-
The difference between the values of y' and y is the inter-
cept, between the two curves, of the ordinate at the point
whose abscissa is a;„ + ^- Its expression is
¥
y.-y.+
[diJ, \di)j' + \\dx')~ tv
1.2+ etc.
Now, consider the case in which the curves meet at the
point P, whose abscissa is a„. Then
182 THE DIFFERENTIAL CALCULUS.
and the intercept of the ordinate will be
which, when Ji becomes infinitesimal, is an infinitesimal of
the first order.
If we also have
the ordinates will differ only by a quantity containing A* as a
factor, and so of the second order. Hence:
Wlien two curves are tangent to each other, they are sepa-
rated only by quantities of at least the second order at an in-
finitesimal distance from the point oftangency.
In the same way it is shown that if the second differential
coefficient also yanishes, the separation will be of the third
order, and so on.
Def. When two curves are tangent to each other, if the
first n differential coefficients for the two curves are equal at
the point of tangency, the curves are said to have contact
of the nth order.
Hence a case of simple tangency is a contact of the first
order. If the second derivatives are also equal, the contact
is of the second order, and so forth.
105. Theorem. In contacts of an even order the two
curves intersect at the point of contact ; in those of an odd
order they do not.
For, in contact of the nth. order, the first term oiy' —y
(§ 104) which does not vanish contains A"+' as a factor.
If n is odd, n-\-l is even, and y' — y has the same alge-
braic sign whether we take h positively or negatively. Hence
the curves do not intersect.
If n is even, m + 1 is odd, and the values oi y' — y have
opposite signs on the two sides of the point of contact, thug
showing that the curves intersect.
CURVATURE; EV0LUTE8 AND INVOLUTES. 183
106. Radius of Curvature. The curvature at any point
is measured thus: We pass from the point P to a point P' in-
finitesimally near it. The
curvature is then measured
by the ratio of the change
in the direction of the tan-
gent (or normal) to the
distance PP'. Let us put
a = the angle which
the tangent at P makes
with the axis of X.
a-\- da = the same angle for the tangent at P'.
ds = the infinitesimal distance PP'-
Then, by definition.
Curvature = -r-.
ds
Flo. 41.
Now, because
we have, by differentiation.
*"^'' = £'
Also,
and
sec' ada = ■-j^,dx.
sec' a—l-\- tan" a = 1 + -^,
ds
= /(
^+%y
From these equations we readily derive
/7/v
Curvature
da
ds
dx'
(^ + g)*
Now, draw normals to the curve at the points P and P',
and let C be their point of intersection. Because they are
perpendicular to the tangents, the angle PCP' between them
will be da, and if we put
184 THE DIFFERENTIAL CALCULUS.
we shall have
pp,
= ds =
■■ pda.
Hence p =
ds
~da
1
~ cuTTature
\ ^ dxV
d'y •
dx'
The length p is called the radius of curvature at the
point P, and C is called the centre of curvature.
CoBOLLAET. The centre of curvature for any point of a
curve is the intersection of
consecutive normals cut-
ting the curve infinitely
near that point. pV
10 T. The Osculating
Circle. If, on the normal
PC to any curve at the
point P, we take any point ^^ ^
as the centre of a circle
through P, that circle will be tangent to the curve at P;
that is, it will, in general, have contact of the first order
at P- But there is one such circle which has contact of a
higher order, namely, that whose centre is at the centre of
curvature. Since this circle will have the same curvature
at P as the curve itself has, it will have contact of at least
the second order at P.
This proposition is rigorously demonstrated by finding that
circle which shall have contact of the second order with the
curve at the point P.
Let us put
X, y, the co-ordinates of P;
jo = ~- for the curve at the point P;
q = -t4 for the curve at the point P,
aUBVATUBE; EV0LUTE8 AND INVOLUTES. 185
These last two quantities are found by difEerentiating the
equation of the curre.
Now, -^ and -^4 must have these same values at the point
dx d^
(x, y) in the case of the circle having contact of the second
order (§ 104).
Let the equation of this circle be
{X - ay + (y- iy = r\ («)
By differentiation, we have
{x — a)dx -\- {y — h)dy = 0,
, dy X — a ,,.
^^"°°" d^'^l^y^P- <*)
DifEerentiating again.
^ - _i_ j_ (^ -a)dy _ {y - by + (r - ay
dx' b-y'^ ip-yy dx (y - dy
r
~{y- by ^'
From (S) combined with (a) we find
{x-ay_
{»)
l+i»= = l +
{y - by (y - by
(i+/r-
r
(y - by
Dividing this by (c) gives
q '
the equivalent of the expression already found for the radius
of curvature.
Hence if we determine a circle by the condition that it
shall have contact of the second order with the curve at the
point P, its radius will be equal to the radius of curvature.
This circle is called the osculating circle for the point P.
Each point of a curve has its osculating circle, determined
by the position, direction and curvature at that point.
186 THE DIFFERENTIAL CALCULUS.
Cor. The osculating circle will, in general, intersect the
curve at the point of contact, for it has contact of the second
order.
This may also be seen by reflecting that the curvature of a
curve is, in general, a continuously varying quantity as we
pass along the curve, and that, at the point of contact, it is
equal to the curvature of the circle. Hence, on one side of
the point of contact, the curvature of the curve is less than
that of the circle, and so the curve passes without the circle;
and on the other side the curvature of the curve is greater,
and thus the curve passes within the circle.
If, however, the curvature should be a maximum or a
minimum at the point of contact, it will either increase on
both sides of this point or diminish on both sides, whence
the circle will not intersect the curve.
108. Radius of Curvature when the Abscissa is not taken
as the Independent Variable. Suppose that, instead of x,
some other variable, u, is regarded as the independent vari-
able. We then have
Now, it has been shown that, in this case, we have (§ 56)
d'y dx d'x dy
d'y _ du' du du' du /„>
\du I
Also, we have
fdyV fdxy fdyV
1 , l^y\'--i , ^(^'"'' _\du) '^\dul .„,
^ + WJ - ^ + Jd^' - my — • ^^^
\du I \du I
These expressions being substituted in the expression for
the radius of curvature, it becomes
CUBVATUBE; EVOLUTES AND INVOLUTES. 187
( \du) + l^j [ ,,,
(4)
cPy dx d'x dy
du' du du' du
109. Radius of Curvature of a Curve referred to Polar
Co-ordinates. Let the equation of the curve be given in the
form
r = (l){d).
The preceding expression (4) may be employed in this case
by taking the angle 6 as the independent variable. By differ-
entiating the expressions
X = r cos 6,
y = r sin 6,
regarding r as a function of 6, we find, when we put, for
brevity,
, dr ,, d'r
r ^ — r zrz
-dd' dd-"
— = — r sm d -\-r' cos 6;
d'x
-^ = (r" — r) cos e — 2r' sin 6;
^—r cos -{- r' sin 6;
^ = (r" - r) sin 6 + 2r' cos 6.
By substituting these derivatives with respect to 6'for those
with respect to u in (4) and performing easy reductions, we
find
_ (r' + r'y _ \ ^ K ddl f
which is the required expression for the radius of curvature.
188 THE DIFFERENTIAL CALCULUB
EXAMPLKS AND EXERCISES.
1. The Parabola. To find the radius of curvature of a
curve at any point, we have to form the value of p from the
equation of the curve. The equation of the parabola is
y' = 2px,
whence we find
dx y'
^ - -t
dx' ~ y''
Then, by substituting in the expression for p, we find
p- f '
the negative sign being omitted, because we have no occasion
to apply any sign to p.
At the vertex y = 0, whence
p=p.
Hence, at the vertex, the radius of curvature is equal to
the semi-parameter, and the centre of curvature is therefore
twice as far from the vertex as the focus is.
2. Show that the radius of curvature at any point {x, y) of
an ellipse is
a*b*
P = „^7.4
and show that at the extremities of the axes it is a third pro-
portional to the semi-axes.
3. Show that the algebraic expression for p is the same in
the case of the hyperbola as in that of the ellipse.
4. What must be the eccentricity of an ellipse that the cen-
tre of curvature for a point at one end of the minor axis may
lie on the other end of that axis? Ans. e = ^/J.
dx
-^— = a — a coBu
du
= y;
d'x dy
-5-, =-r- = asiau;
du' du
d'y
— = a cos u.
OURVATUBE; EV0LUTE8 AND INVOLUTES. 189
5. Show that in the case supposed in the last problem the
radius of curvature at an end of the major axis will be one
fourth that axis.
6. The Cycloid. By differentiating the equations (1), § 80,
of the cycloid, we find
(3)
Then, by substituting in (4) and reducing, we find, for the
radius of curvature,
p ■= 2*a Vl — cos w = 4a! sin \u.
"We see that at the cusp, 0, of the cycloid, where m = 0,
the radius of curvature also becomes zero.
7. The Archimedean Spiral. Show from (5) that the ra-
dius of curvature of this spiral (r = aO) is
8. The Logarithmic Spiral. The equation of the loga-
rithmic spiral being
w
r = ae ,
show that the radius of curvature is
p = r vr+f.
Hence show that the line drawn from the centre of curva-
ture of any point P of the spiral to the pole is perpendicular
to the radius vector of the point P-
9. Show that the radius of curvature of the lemniscate in
terms of polar co-ordinates is
a a'
P =
3 Vcos %e 3r-
190
THE DUVEBENTIAL CALCXTLUS.
no. Evolutes and Involutes. For every point of a curve
there is a centre of curvature, found by the preceding for-
mulae. The locuB of all such
centres is called the evolute
of the curve.
To find the evolute of a
curve, let {x,y^) be the co-ordi-
nates of any point P of the
curve ; PO, the radius of cur-
vature for this point; and a,
the angle which the tangent
at P makes with the axis of X.
Then, for the co-ordinates of
0, we have
X =^ Xj — p sm a;
y = yj + P0OB a.
Substituting for p its value (§ 106), and for sin a and cos a
their values from the equation
_^
dxf
we find
Fio. 48.
tan a ■■
«> —
1 + ^'
dx' dy,^
= 2// 4
dx;
^ dx;
dx;
dx.
(1)
If in the second members of these equations we substitute
the values of the derivatives obtained from the equation of
the curve, we shall have two equations between the four vari-
ables X, y, x^ and y^. By eliminating x^ and y from these
equations and that of the given curve, we shall have a single
equation between x and y, which will be that of the evolute.
CUBVATUBBj EVOLUTES AND INVOLUTES. 191
111. Case of an Auxiliary Variaile. If the equation of
the curve is expressed by an auxiliary variable, we have to make
in (1) the same substitution of the values of -^', -r^./, etc, ,
as in § 108. Thus we find, instead of (1),
fdxX Idy^
dy^ \dul \du
du d'y, dx, d'x, dy/
du' du du' du
IdxX IdyX
dx, \du I \du I
which are the equations of the evolute in the same form.
du d'y^ dx, d'x, dy^ '
du^ du du' du
(3)
EXAMPLES OF EVOLUTES.
113. The Evolute of the Parabola. If we substitute in
(1) for the derivatives of y, with respect to x, the values
already found for the parabola, these equations (1) become
, V/' ,3 y;
' -^ p ^ ill p
" p
We now have to eliminate y^ from these two equations, x^
having already been eliminated by the equation of the curve.
They give
y/ = M^-p)> y/ = -fy-
Equating the cube of the first equation to the square of the
second, we find, for the equation of the evolute of the parabola.
y =
8 (x - pY
27
P
192
THE BIFPBBBNTIAL GALOULUS.
113. Evolute of the Ellipse. Prom the equation of the
ellipse, we find
dx^ ai'y/ dx/ o^y''
By substituting in (1) and reducing, we find
Eemarking that the equation of the ellipse gives
a'V - a*y; = a'5V,
and putting e' = a' — S",
the preceding equation becomes
X = -^. (a)
In the same way we get
%
In this case the easiest way to effect the elimination of x,
and y^ is to obtain the values of these quantities from («)
and (5), and then substitute them in the equation of the
ellipse. Prom (a) and (p), we find
which values are to be
substituted in the equa-
tion
1.
^^
1-
■^\^
/C.y
\
V^B P \
V N
/
-^ V
We thus find, for the
equation of the evolute of
the ellipse,
ffM + b^y^ = c*.
The figure shows the ^"^ ^*-
form of the curve. The following properties should be de-
duced by the student.
OU^VATUBE; EVOLUTES AND IN70LUTE8. 193
(a) The evolute lies wholly within the ellipse, or cuts it (as
in the figure), according as e' < ^ or e' > \.
ifi) The ratio CD : AB (which lines we may call axes of
the evolute) is the inverse of the ratio of the corresponding
axes of the ellipse.
114. Evolute of the Cycloid. Here we have to apply the
formulae (3) for the case of a separate independent variable.
Substituting in (2) the values of the derivatives already given
for the cycloid, we shall find
IdxV
\dul
d^y dx
du' du
d'x dy _
cos u);
— cos u);
du'
x= Xi-\-%a sin u = a{u -\- sin u);
y ^= y, — 2a(l — cos u) = — a{l — cos u).
These last two equations are those of the evolute.
Let us investigate its form. For m = we have x =
and y — 0, whence the
origin is a point of the
curve.
For u = TT we have
^
Y
c
Fio.
X
45.
y=-%a;
giving a point C, below
the middle of the base of
the cycloid, at the dis-
tance 2a. Let us take this point as a new origin, and call
the co-ordinates referred to it x' and y' . We then have
x' ^^ X — an =■ a(d — n -\- sin ff);
y' — y -\-%a = a(l + cos ff).
If we now put
these equations become
194
THE DIFFERENTIAL CALCULUS.
x' = a{0' - sin (9');
y' = a{l - cos 6');
which are the equations of another cycloid, equal to the
original one, and similarly situated. The cycloid therefore
posesses the remarkable property of being identical in form
with its own evolute.
115. Fundamental Properties of the Evolute.
Theorem I. The involute of a curve is the envelope of its
normals.
As we moTe along a curve, the normal will be a straight
line moving according to a certain law depending upon the
form of the curve. This line will, in general, have an en-
velope, which envelope will be, by definition, the locus of the
point of intersection of consecutive normals. But this point
has been shown to be the centre of curvature, whose locus is,
by definition, the evolute. p.
Hence follows the theorem.
OoKOLLAET. The nor-
mals to a curve are tan-
gents to its evolute. For
this has been shown to be
true of a moving line and
its envelope.
Theokem II. If the os-
culating circle move around
the curve, the motion of its
centre is along the line join-
ing that centre to the point
of contact.
This theorem will be
made evident by a study
of the figure. If the line
F^C, be one of the nor-
mals from the point of contact P, to the centre, then, since
Fio. 46.
CUMVATUBB ; EVOLUTES ANB INVOLUTES. 195
this normal is tangent to the locus of the centre, it will be the
line along which the centre is moving at the' instant.
Theobem III. The arc of the evolute contained between
any two points is equal to the difference of the radii of the
osculating circles whose centres are at these points.
For, if we suppose the points C,, 0^, etc., to approach in-
finitesimally near each other, then, since the infinitesimal
arcs C^C^, 0,0^, etc., are coincident with those successive
radii of the osculating circle which are normal to the curve,
these radii are continually diminished by these same infini-
tesimal amounts.
The analytic proof of Theorems II. and III. is as follows:
Let the equation of the osculating circle be
{X - ay + {y-~ iy = p\
where a and 5 are the co-ordinates of the centre of curvature,
and therefore of a point of the evolute.
The complete differential of this equation gives
{x — a) {dx — da) -\- (jy — i) {dy — db) = pdp. (a)
If, in this equation, we suppose x and y to be the co-ordi-
nates of the point of contact of the circle with the curve, then
dx and dy will have the same value at this point whether we
conceive them to belong to the circle, supposed for the mo-
ment to be fixed, or to the curve. But in the fixed circle we
have
{x — a)dx + (y - b)dy = 0. (5)
Subtracting this equation from (a) and dividing by p, we find
da + db = — dp, (c)
p p
which is a relation between the differential of the co-ordi-
nates of the centre and the differential of the radius. Now,
if we put /3 for the angle which the normal radius makes
with the axis of X, we have
X — a y — a .
= cos yS; = sm p. (d)
196 THE DIFFERENTIAL CALCULUS.
But this same normal radius is a tangent to the evolute.
If we call <T the arc of the evolute, we find by a simple con-
struction da = cos I3d(r; db = sin ftd(T,
Multiplying these equations by cos /S and sin C, respectively,
and adding, we find
d(T = cos /3da + sin /3dl>.
Comparing (c) and (d), we find
d(T =■ — dp,
or d((T + /o) = 0.
Now, a quantity whose differential is zero is a constant.
Hence we always have
a -\- p = constant,
or cr = constant — p.
If we represent by a-, and o", the arcs from any arbitrary
point of the involute to the two chosen points, and by /a, and
p, the values of p for these points, we have
<r, = const. — p,;
<7, = const. — Pj.
.'. 0-, - 0-, =P, - A>
or the intercepted arc equal to the difference of the radii, as
was to be proved.
It must be remarked, however, that whenever we pass a
cusp on the evolute, we must regard the arc as negative on
one side and positive on the other. In the case of the ellipse,
for example, those radii will be equal which terminate at
equal distances on the two sides of any cusp, as A, B, or
D, and the intercepted arc must then be taken as zero.
116. Involutes. The involute of a curve O is that
curve which has C as its evolute.
The fundamental property of the involute is this: The
involute may be formed from the evolute by rolling a tangent
OUBVATUBOEj EV0LUTE8 AND INVOLUTES. 197
line upon the latter, A point P on the rolling tangent will
then describe the involute.
This will be seen by reference to Fig. 46. The rolling line,
being tangent to the evolute, coincides with the radius P^G^,
and as it rolls along the evolute into successive positions,
P,(7„ P,C„ etc., the motion of the point P is continually
normal to its direction.
It will also be seen that the radius of curvature of the in-
volute at each point is equal to the distance PC from P to
the point of contact with the evolute.
The conception may be made clearer by conceiving the
rolling line to be represented by a string which is wrapped
around the evolute. The involute is then formed by the mo-
tion of a point on the string.
The general method of determining the involutes of given
curves involves the integral calculus.
PART II.
THE INTEGRAL CALCULUS.
PART II.
THE INTEGRAL CALCULUS.
CHAPTER I.
THE ELEMENTARY FORMS OF INTEGRATION,
117. Bejinition of Integration, Whenever we have given
a function of a variable x, eay
M = F{x),
we may, by differentiation, obtain another function of x,
which we call the derimd function.
In the integral calculus we consider the reverse process.
"We have given a derived function
F'{x),
and the problem is: What function or functions, when differ-
entiated, will have F'{x) as their derivative?
Every such function is called an integral of F'{x),
The process of finding the integral is called integration.
The operation of integration is indicated by the sign / ,
called " integral of," written before the product of the given
function by the differential of the variable. Thus the ex-
pression
' F'{x)dx
P
-means: that function whose differential with respect to x is
F'(x)dx.
202 THE INTEGBAL CALCULUS.
Calling M the required function, then if we have
we must also have
u = fF'{x)dx=f^dx.
As examples:
Because d{x') = %xdx,
we have / ^xdx = x'.
Because dijxx' ■\-hx-\-c) = {2ax + i)dx,
we have / {2ax -\-b)dx = ax^ -{- bx-{-c.
And, in general, if, by differentiation, we have
dF{x) = F'{x)dx,
we shall have / F'{x)dx = F{x).
118. Arbitrary Constant of Integration. The following
principle is a fundamental one of the integral calculus:
If F{x) is the integral of any derived function of the va-
riable X, then every function of the form
F{x) + h,
h being any quantity whatever independent of x, will also be
an integral.
This follows immediately from the fact that h wUl dis-
appear in differentiation, so that the two functions
F(x) and F{x) + h
have the same derivative (cf. § 24).
The same principle may be seen from another point of
view : Since the problem of differentiation is to find a func-
tion which, being differentiated, will give a certain result,
and since any quantity independent of the variable which
may be added to the original function will have disappeared
by differentiation, it follows that we must, to have the most
TEM ELEMBNTABT FORMS OV INTEGRATION. 203
general expression for the integral, add this possible but un-
known quantity to the integral.
The quantity thus added is called an arbitrary constant.
But it must be well understood that the word constant merely
means independent of the variable with reference to ■ which
the integration is performed.
It follows from all this that the integral can neyer be com-
pletely found from the differential equation alone, but that
some other datum is needed to determine the arbitrary con-
stant and thus to complete the solution.
Such a datum is the value of the integral for some one
value of the variable. Let F{x) -\-. h be the integral, and let
it be given that
when X = a, then the integral = K.
"We must have, by this datum,
F(a) + h=K,
which gives h = K — F{a),
and thus determines h.
Eemaek. Any symbol may be taken to represent the ar-
bitrary constant. The letters c and h are those most gener-
ally used. We may affix to it either the positive or the nega-
tive sign, and may represent it by any function of arbitrary
but constant quantities which we find it convenient to intro-
duce. It is often advantageous to write it as a quantity of the
same kind as the variable which is integrated.
119. Integration of Entire Functions.
Theokem: I. The integral of any power of a variable is
the poioer higher by unity, divided by the increased exponent.
In symbolic language, we have
/j-n + l
x'dx = — —r + h,
n+1
jpn + l
For, by differentiating the expression — — - -\- h, we have
^ w + 1
204 THE INTEGRAL CALCULUa.
Theorem II. Any constant factor of the given differen-
tial may be written before the sign of integration.
In symbolic language.
J'aF'{x)dx = aJ'F'{z)dx
This is the converse of the Theorem of § 33. By that
theorem we have
d{aF{x)) = adF{x),
from which the above converse theorem at once follows.
In the special case « = — 1 we have
y*- F'{x)dx = fF\x)d{- x) = - jF'{x)dx.
Hence the corollary: If the integral is preceded by the nega-
tive sign we may place that sign before either the derived
function or the differential.,
Theoeem III. If the derived function is a sum of several
terms, the integral is the sum of the separate integrals of the
terms.
In symbolic language,
y(X+ r+ Z+ . . .)dx = J Xdx-\- J Ydx-\- f Zdx^ . .
This, again, is the converse of Theorem II of § 23.
The foregoing theorems will enable us to find the integral
of any entire function of a variable. To take the function in
its most general form, let it be required to find the integral
u= I {ax'" + Sa;" -\- ex" -\- . . .)dx.
By Theorem III.,
U= j as^dx -\- I bafdx + / C3^dx -f , o . .
THE ELEMENTARY WORMS OF INTEGRATION. 205
By Theorem II.,
/ ax^dx = a I x'^dx;
etc. etc. ;
and by Theorem I.,
/
jgm + l
x'^dx = — — j- + A,;
etc. etc.
By successive substitution we then have
where 7t„ 7^„ h„ etc., are the arbitrary constants added to tha
separate integrals.
Since the sum of the products of any number of constants
by constant factors is itself a constant, we may represent the
sum aA, + 57t, + ch^ by the single symbol li. Thus we have
jiax'" + 5a;" + ca;" + . . .)dx
_ ««:'»+ ' Sa;"+' , ca;^+'
EXERCISES.
Form the integrals of the following expressions, multiplied
by dx:
I. x'. 2. x'. 3. a;"'. 4. «"".
5. ax*. 6. Ja;°. 7. ax~\ 8. Sa;~^
9. fta; + 5. 10. ax' — c. 11. aa;' + ca;. 12. ax' — co^
13. a:*. 14. a;». 15. x-i. 16. aa;-*.
w ah ,1
17. aa:*— Ja;-*. 18. wa;* — — . 19. —,——,. 20. a-\ — ,.
130. The Logarithmic Function, An exceptional case
of Theorem I. occurs when w = — 1, because then n -\-l
= 0, and the function becomes infinite in form. But since
d'los: X = — = x~*dx,
° X
(*)
206 THE INTEGRAL CALCULUS.
it follows that we haye for this special case
J x~^dx = / — = log a; + 7i. (a)
Let c be the number of which /* is the logarithm. We then
have
log a; + A = log a; + log c = log ex.
We may equally suppose
A = — log c = log -.
c
Then log a; + A = log -.
c
Hence we may write either
rdx - 1
y - = log ex,
/dx , x
— = log -;
X ° c
c being an arbitrary constant.
We thus haye the principle: The arbitrary constant added
to a logarithm may be introduced by multiplying or dividing
by an arbitrary constant the number whose logarithm is ex-
pressed.
131. We may derive the integral (a) directly from Theo-
rem I., thus: In the general form
/j.n + 1
x'dx = — r-5- + h
let us determine the constant h by the condition that the in-
tegral shall vanish when x has some determinate value a.
This gives
— -— + /t = 0; .-. h= 1-^.
n -\-l n-\-l
Thus the integral will become
/ x^dx = — ,
TEE ELEMENTAMT FORMS OF INTE&BATION. 207
in which a takes the place of the arbitrary constant. This
expression becomes indeterminate for to = — 1. But in this
case its limit is found by § 71, Ex. 5, to be log x — log a.
Thus we have
/
2/
x^dx = log X — log a = log — ,
as before, log a being now the arbitrary constant,
133. Exponential Functions. Since we have
d{d') = log a-a^dx,
it follows that we have
I log a. a'dx =.0," -J- Ti,
or, applying Th. II., § 119, to the first member and then di-
viding by log a,
d'-\-h
/.
log a
which we may write in the form
/
a^dx — = f- h,
log a '
because = is itself a constant which we may represent by h.
133. The Elementary Forms of Integration. There is
no general method for finding the integral of a given differen-
tial. What we have to do, when possible, is to reduce the
differential to some form in which we can recognize it as tha
differential of a known function. For this purpose the fol-
lowing elementary forms, derived by diSerentiation, should
be well memorized by the student. We first write the prin-
cipal known differentials, and to the left give the integral,
found by reversing the process. For perspicuity we repeat
the forms alrea,dy found, and we omit the constants of in-
tegration.
20S08
THE INTEGRAL OALCULUS.
.•^(y + >) ={n + l)y''dy,
.-d-logy =-^,
.• d'siny = cos ydy,
. • tZ" cos y = — sin ydy,
.•d'tany = aec' ydy.
d'cot y
_ ^y
sm' y'
, tan y J
.■a-sec« = -dy,
" cos y ^
• <Z"sin<~'' 2/ =
, • fZ-cos'~"2/ =
<Zy
1/1-2/'
dy
Vi-f'
..^.tan<-).2/=i^-„
. • J-smh<~"2/=
a" log ady,
dy
fray =1^;. (1)
/| =log2/. (2)
/ COS ydy = sin y. (3)
/ sin ydy = — cos y. (4)
tan y. (5)
- cot y. (6)
r dy ^
J cos' y
r dy
J sin'
sm ^
/tan y<?y
cos y
r -dy _
= sec y. (7)
sin<-"'y. (8)
cos<-'>2/. (9)
«/ 4/1-2/'
.Ja^dy
log a'
• (11)
V/+1
. • .f-r^ = Sin li<-'>2^ = log (y + Vf+1), (12)
<f-COSh<-"«= — ;==
V2^'-
/
■ d-isinM~^^y=
dy
= COS h<-«2/ = log {y + 4/2/'- 1). (13)
1-2/"
■••/r^,-' =tanh<-V = ^log^^.
(14)
INTBQItALS BEDUOIBLE TO ELEMENTARY F0SM8. 209
CHAPTER II.
INTEGRALS IMMEDIATELY REDUCIBLE TO THE
ELEMENTARY FORMS.
134. Integrals Reducible to the Form I y'dy. The fol-
lowing are examples of how, by suitable transformations, we
may reduce integrals to the form (1). Let it be required to find
I (a-\- xYdx,
We might develop (a + a;)" by the binomial thorem, and
then integrate each term separately by applying Theorem III.,
§ 119. But the following is a simpler way. Since we have
dx — d{a + x), we may write the integral thus:
Ha + xYd{a + x).
It is now in the form (1), y being replaced by a -j- a;.
Hence
J^a + xYdx=^^±^^-+h. (1)
In the same way,
f{a - xYdx = - f{a - xYd{a - x) = h - (" ~f^" —.
To take another step, let us have to find
/"{a + bxYdx.
We have
dx = j-d(lix) = Td{a -\- hx). ,
Hence, by applying Th. II.,
fia+bxYdx=^/(a+hxYd{a+bx)= i^±M.^ + j,^ (3)
210 THE INTEGRAL CALCULUS.
We might also introduce a new symbol, y ^a-{-hx, and
then we should have to integrate y'^dy with the result in § 123.
Substituting for y its value in terms of x, we should then have
the result (2).*
These transformations apply equally whether n, a and b
are entire or fractional, positive or negative.
EXERCISES.
Find: i. j{a + xydx. 2. y 3(a — xydx.
2,. I {a — 2xydx. 4. / (a + x)-*dx. 5. I (a — x)-^dx.
I {a-\-mx)~'dx. "]. I {a — mxfdx. 8. / {a — mo^~^dx.
/dx n dx f dx
(a + xY' '°V (a - xY' ^^'J I
IS
17
18,
19
(« + «)'• J {a-xY' J{a-^xy
I {a'\- xydx. 13. / (a + nxydx. 14. / (a + x'yxdx.
«/ \ »' ' a; a; V t/ (a — a;)"
^ IC^"^^' + (a - a:)' + (a - a;)V '^'^'
t' \(a — mxy {a — mxy {a — mxy)
I (a-\-lx-\- cx'){b + 2cx)dx.
f{a + I)X + cafyib + 2cx)dx.
r {b-\-2cx)dx
J (a + bx -\- cx'y
* The question whether to introduce a new symbol for a function
whose differential 'is to be used must be decided by the student in each
case. He is advised, as a rule, to first use the function, because he then
gets a clearer view of the nature of the transformation. He can then
replace the function by a new symbol whenever the labor of repeatedly
writing the function will thereby be saved.
INTBGRALa BEBUCIBLE TO ELEMENTABT FORMS. 211
135. Application to the Case of a Falling Body. We
have shown (§ 33) that if, at a time t, a body is at a distance
z from a point, the velocity of motion of the body is equal to
dz
the derivative -:-. Now, when a body falls from a height
under the influence of a uniform force g of gravity, unmodi-
fied by any resistance, the law in question asserts that equal
velocities are added in equal times. That is, if z be the
height of the body above the surface of the earth, and if we
count the time t from the moment at which the body began
to fall, the law asserts that
dz . . .
the negative sign indicating that the force g acts so as to
diminish the height z.
By integrating this expression, we have
z:=zh- igf. (5 )
Here the constant h represents the height z of the body at
the moment when ^ = 0, or when the body began to fall.
From the definition of h and z, it follows that 7j — 2 is the
distance through which the body has fallen. The equation
(b) gives
h — z = igt". (c)
Hence: The distance through which the tody has fallen is
jiropiortional to the square of the time.
At the end of the time t the velocity of the body, meas-
ured downwards, is, by (a), equal to gt. If at this moment
the velocity became constant, the body would, in another
equal interval t, move through the space gt X t = gf.
Hence, by comparing with (c) we reach by another method a
result of § 33, namely:
hi any period of time a lody falls from a state of rest
through half the distance through which tt would move in the
same period with its acquired velocity at the end of the period.
212 THE INTEGRAL GALCUZm.
136. Reduction to the Logarithmic Form. Let us have
to find
/mdx
ax-\-o
Since dx = —d{ax) = —d(ax + 5)>
we may write this expression in the form
/m d{ax + S)
a ax-\-b '
and the integral becomes
m pdiax -\-V\ m , ax + h
u= - -^ ~jf = - log —-,
aJax-\-o a ° c
e being an arbitrary constant.
EXERCISES.
Integrate the following expressions multiplied by dx:
,1 b m
I. X + -. 2. -. 3, -.
X X X
1 ^ wi
o.
x + r ^' 2x- r
ex —
_m^ „ ffl_ a'
'• ^ + '^ • 2ax + b' ^' %bx + a''
3? -^Jcx a-\-b m — n
10. — ; — ; — i— . II. ; 7. 12. .
4 + « ooj+o mx — n
^. ^ n xdx p xdx
Note that xdx = ld{x') = i<Z(l + x').
p xdx ,/•»;' dx /'log a; <?»
Note that log x— =^\ogxd. log a;.
/»loff(l + w), p xdx n xdx
INTEGRALS BEDUCIBLE TO ELBMENTABY F0BM8. 213
I'/iT. Trigonometric Forms. The following are examples
of the reduction of certain trigonometric forms:
/cos mx dz ■= — I cos mx dUnx) = — sin mx + h.
mj ^ ' m
/sin mx dx = — I sin mx d(mx) = h cos mx.
mJ ^ ' m
I cos (a -\- mx)dx = — / cos (a -f- mx)d{a -j- mx)
_ sin (a + ?nx)
— -j- /I.
m
/, -, /»siD.xdx nd'cosx
tan xdx= I = — /
J cos X J cos X
= h — log cos X — log c sec x,
where h = log c.
In the same way,
/ cot xdx = log c sin a;.
/dx n 1 dx pd'tan x ,
-. = / 7 = / -T — = log c tan X.
sm X cos a; J tan x cos a; ^ tan z °
/dx 1 p dx 1.1
-: — = o / -• — i 7- = log c tan -X.
sm a; 2 J sm -Ja; cos -Ja; ° 2
/ dx _ n dx _ ^ /;r a; \
cos x~ J sin (i;r — a;) ~ ^ \4 2 /"
EXERCISES.
Integrate :
I. (1 + cos y)dy. 2. (1 — e sin u)du.
3. cos 2y dy. Ans. i I cos 2yd{2y) = i sin 2y.
4. sin 2y dy. 5. cos my dy.
6. sin y cos y dy. Ans. i I sin 2yd{2y) = — J cos 2y.
7. tan 2a; dx. 8. cot 2a: dz.
9. 2 cos' X dz.
Ans. J 0- + cos 2z)dz = a; + A -f ^ sin 2a;.
214 THE INTEGRAL CALCULUS.
lo. 2 sin* xdz. 1 1. tan 3y dy.
cos « (iv , rdi'i- + sin «) , /i , • \
1 + sm y J l-\-smy o v .
sin tf <?« sin y dy
■^ 1 + cos y
cos y dy ,
15. q -V-^. 16.
1 — sm y
cos' a; — sin' x .
17. ; — J, oa;. 18.
sm %x
dx dx
19. . 20.
1 — cos y'
sin 3y dy
cos 2/
sin 2a;
rJr
cos' X — sin'
X
21. -: —
dx
cos mx sin ma; sm mx cos ma;
22. sin [mx + o)<?a;. 23. cos (a — nx)dx.
24. tan nx dx. 25. tan (2a; — a)dx.
^ dx dx dx
20. -; -, r. 27. TT r. 26. —. J r.
sm (a — x) cos (0 — ma;) sm (a — nx)
cos »i«(i!/ sin nydy sec' a;t?a;
20. ;^ -^ . -50. -■^— ^. •51. T .
^ a-{-s\D.ny a — cos ny a— m tan a;
138. Integration of -i— — j awtZ -; ;.
(Z ~T* 3/ CI — X
We see at once that the first difEerential may he reduced to
that of an inverse tangent ; thus,
dx 1 dx
i4)
Hence
x' + a' a'x' , ^ ax' '
a' + ^ J' + ^
d-
/'_^_ = L r-JL, = L tan<- « i +
J a' + ay" a'^ £ , 1 « <^
«' +
(1)
"We find in the same way
/'_,^ = ltanh<-)^ + A = llog«^±^, (2)
,/ a' — x' a a %a '^ a — z ^
c being an arbitrary constant factor.
INTEGRALS REDUGIBLB TO ELEMENTABT FORMS. 215
dx
139.
Integrals of the form / —
+ bx -\- ex''
The reduction of integrals of this form depends upon the
character of the roots of the quadratic equation
cx' + bx+a = 0. (1)
I. If these roots are imaginary, the integral is the inverse
of a trigonometric tangent.
II. If the roots are real and unequal, the integral is the
inverse of an hyperbolic tangent.
III. If the roots are real and equal, that is, if the above ex-
pression is a perfect square, the integral is an algebraic frac-
tion.
Dividing the denominator of the fraction by the coefficient
of x', the given integral may be written
dx
\f
, ^ bx , a
c c
(«)
Writing 2p for — and q for — , the expression to be inte-
c c
grated may be reduced to one of the forms of § 128, thus:
dx dx d{x -\-p)
x' + 2px"+q ~ (^ +!>)' + ~q-f~{x +pY +'q^'' ^'
The three cases now depend on the sign oi q — jo'.
I. If q—p' is positive, the roots of (1) are imaginary and
the form is the first of the last article with x-\-p in the place
of X, and g' — ^' in the place of a'. Hence we have
x' 4- %px +~q ~ J {x + pY + q-p'
= , ^ ^ tan<-') f +^ +A. (1)
Vq -y Vq-p' ^ '
Comparing this result with (a), we see that this integral
may be reduced to its primitive form by changing p into
216 THE INTEGRAL CALCULUS.
5- — and q into — . Substituting and reducing, we have
dx 1 p dx
/dx _ 1 /"
a -\- bx ■\- cx^ ~ c J ,
x' +-X + -
c c
x + ^
:l._A_tan<-» ^^«
c
'^ c 4c' '^ c 4c'
; tan< *' — + h. (3)
V4ac - /^' ¥4:00 - b'
II. If g —p' IB negative, that is, if 4ac — b' is negative in
(3), the expression (2) will contain two imaginary quantities.
But these two quantities cancel each other, so that the ex-
pression is always real. When q —p' is negative, we write
(b) in the form
The integral is now in the form (2) of § 128, and we have
tfa; _ /» d{x + p)
/ux _ p a(x
x" + 2px + q~ J p' — q —
: tanh(-«
x+p
Vp' — q Vp' — q
-* Vp — q yp— i—Kp-Vp)
Making the same substitutions in these equations that we
made in Case I., we find
r dx ^ ^ _ a tanh<-» J^±l.
J a + bx + cx' Vb' — 4:ac Vb" - iac
1 , Vb'-^ac + 2cx + h ,,,
= h ; log c , ■'—•(4)
4^*'-4ac 4/6''-4ac-(2ca;+5)
III. If ^' — g' = 0, the expression to be integrated becomes
dx
We have already integrated this form and found
/dx , 1
(-— = *-
INTMaBALS BEDUOIBLE TO ELEMENTARY FOBMS. 217
EXERCISES.
Integrate the following expressions:
dx. dx dx
I.
x' -2x- 4'
dx
" («-«)(»■
dx
-/?)•
^' a4-2Ja;-a;''
dx
(a — x){x —b)
x' + Ax + 2'
"' x{x — a)'
130. Inverse Sines and Cosines as Integrals. From what
has abeady been shown (§ 123, (8) and (9)), it will be seen that
we have the two following integral forms:
— = sin <-" a; + A = u; (a)
f -^— = ecu (- ') ^ + h.' := 7//; (h)
where we have added h and h' as arbitrary constants of in-
tegration.
Comparing the first members of these equations, we see
that each is the negative of the other. The question may
therefore be asked why we should not write the second
equation in the form
u' = - f— r^— = A" - sin <- « X, (c)
as well as in the form {h). The answer is that no error
would arise in doing so, because the forms {b) and (c) are
equivalent. Trom {V) we derive
X = cos (m' — A') = cos (h' — u'); (d)
and from (c),
X = sin (A" — u'). (e)
Now, we always have sin (a + 90°) = cos a. Hence {d)
and («) become identical by putting
h" =:h' + 90°,
which we may always do, because the value of h" is quite
arbitrary.
218 THE INTEGRAL CALCULUS.
131. The preceding reasoning illustrates the fact that
integrals expressed by circular functions may be expressed
either in the direct or inverse form. That is, if the relation
between the differentials of u and x is expressed in the form
- dx
du =
Vl - x"
we may express the relation between u and x themselves
either in the form
u = sin^~"a; + A
or in the form x = sin (m — h).
So, also, in the form (1) of § 138 we may express the rela-
tion between x and u either as it is there written or in the
reverse form,
X = a tan a{u — h).
dx
133. Integration of ■
We have
d--
(1)
In the same way
/'--==^ = cos<-»- + A or A -sin'-"-. (3)
•^ Va' — x' « a ^ '
We also have
d--
/• ^^ -= /•-^J= = sinh'-«i + A
= log -(a; + 4^^+^). (3)
INTEGRALS REDUCIBLE TO ELEMENTARY FORMS. 219
d--
J Vz' -a' J J^ _ a
= log -(« + V*' - a'). (4)
a
Integrate the differentials:
dx
2
Vc-x'
ndy
•^ Va' - nY
4'
mdz
e —
6
" V^a' - m'z'
dx
8.
Vic' + x'
dy
9- ,. .\ -T--
lO,
dy
Via
dx
Vcf
-{X-
dz
ay
Via
' - m'z'
mdx
V7^
+ mV
dx
V^
+ m'{x
dx
-ay
V(i"
-ay-
•4c"
«a;"
-^dx
V'4a' + %'
dy
II. , 12.
Vl + c{x - ay
2xdx
13. — - . 14.
Va' - z* Va"" - x""
-. — cos xdx
15. If du = —r======^ then sin a; = a cos (u -{■ ,
\ (t ~~" SlU iC
e'dx dx
16. , 17. — , -.
Vl -e"^ ^Vl- (log xy
„ — sin Kifo cos xdx
1 8- ::i-r^:7r.--- i9.
a' + cos' a;' ^ «' + sin" x
(x — a)(5?a; (x 4- a)dx
20. -^ 21.
Vl-(x-ay Vl + {x + ay
220 THE INTEGRAL CALCULUS.
133. Integration of Every difEerential of
Va-\-bx ± ex'
this form can be reduced to one of the three forms of the
preceding article by a process similar to that of § 129. The
mode of reduction will depend upon the sign of the term ex'.
Case I. The term ex' is negative. Putting, as before,
16 a
P^%-e' 'i = -e'
we have
^a-\-'bx — ex' = Vc Vq-\- 2px — x' = Vc Vp' + q— (x—p)'.
Then, comparing with (1) of § 132, we find
r dx _ J_ /• d{x—p)
'^ Va+'bx - ex' ~ Vc'^ Vf + q- (x-p)'
= —— S1T1 < '' , ■^ — = — — R1T1 ' '>^ — (1)
V^ ^p' + q Vc V¥+lac
In order that this expression may be real, p' -{■ q qt b' -\- iac
must be positive. If this quantity is negative the integral
will be wholly imaginary, but may be reduced to an inverse
hyperbolic sine multiplied by the imaginary unit.
Case II. The term ex' is positive. We now have
Va + bx+ ex' = Vc V{x + py + q —p'-
/dx _ 1 /" ^(.^ + P)
Va + bx+ ex' ~ VeJ V(z+py + q-p'
= —. log G{x-\-p^ i/x' + 2px + q)
Vc
= -,- log ^2cx + b + 2c* Va + bx+ ex').
Because Cis an arbitrary quantity, the quotient of by
2c3 is equally an arbitrary quantity, and may be represented
by the single symbol O. Thus we have
dx 1
/■
Va-\-bx-\- ex' c*
log 0{b+2cx+2 Vc Va+bx+cx').{2)
INTE0BAL8 REDUCIBLE TO ELEMENTARY FORMS. 221
EXERCISES.
Integrate:
dy ^ dy
Via' + Uy - f V(a + y){b- y)
ydy dy
V8 - iy' + y*' ' VaY - by + V'
cos ddd , cos 6d0
S- -7=======.. 6.
V\ - sin 6* — sin" d Vl - sin ^ + cos"
sin 6 cos ddd a sin Bdd
o.
V4 - cos 26/ - cos'' %e Va' - b\l - cos 0)'
134. Exponential Forms. Using the form (11) of § 133,
we may reduce and integrate the simplest exponential dif-
ferentials as follows:
/I /» a""
a'^dx = - / a'^dOmx) = -^, \- h. (1)
mj ^ mloga ^ '
fa'+'-dx = fa'+"d{x + b)= ^ + h. (2)
/I „ „mx+b
a'^ + ''dx=- /a'^ + '>d(mx + b)=—^ \- h. (3)
mJ ^ ' ' mloga ^ '
a-'-^dx = - - / a-'^d(- mx) = -^^-^ . (4)
mJ ^ I . mlog a ^ '
EXERCISES.
Integrate:
I. tf°dx. 2. b^dy. 3. a''~^dy.
4. (o + J)e^<fa;. 5. a^-'dy. 6. a-^dx.
7. (a'' + a-'')«Za;. 8. (a'" — a-'")(Za;. 9. (a + e==)<Za;.
a'"— a '^)dx. II. r — j — -. 12.
1 + e"'" * 1 + e"^*
'3- gx^e- -^- 14. (1 + a'Ydx.
15. (a"« + a-'^)'<Za:. 16. f'e''''xdx.
17. Cef^'xdx. 18. re-<>^'''-'^)xdx.
222 THE INTEORAL CALCULUB.
CHAPTER III.
INTEGRATION BY RATIONAL TRANSFORMATIONS.
135. We have now to consider certain forms which cannot
be reduced so simply and directly as those treated in the last
chapter. Before passing to general methods we shall consider
some simple cases.
I. Integration of ^ s-^rf«. Any form of this kind, when
m is entire, may be integrated by developing the numerator
by the binomial theorem. We then have
{a + a;)"* _ £^ , wa^-'a ; (m\ a''-'x '
and each term can be integrated separately. If w < »i + 2,
and entire, one of the terms of the integral will contain log x.
II. Integration of . , -. We may reduce this form to
the preceding, by introducing a new variable, 2, defined by
the equation
e = a -\- ix.
mi,- • z — a J dz
This gives X = — T — ; ax = -r.
Substituting these values of z and dx in the expression to
be integrated, it 'becomes
{z — a)'"dz
Jm + l^i. J
which may be integrated by the method of the last article.
xdx
III. Integration of — r-^ — — — 5. We reduce the denomi-
INTMOBATION BY RATIONAL TBAN8P0BMATI0N8. 223
nator to the form ± {f — q) ± {x+pY a&ia% 139. Then,
putting, for brevity,
V =f - q.
z = x+p,
which gives dx — dz,
the integration will have to be performed on an expression of
the form
b' ± z'~ ~ W~±? ~ F±l"''
Each of these terms may be integrated by methods already
given (§§126, 128).
The process is exactly the same if we have to find
(a + bx)dx
/(a -f- oxy
pY
EXERCISES.
Integrate: /I 1\*
{x — aydx
\a ^1
dx
X' X
x'dx x'dx
•'• (a - xy '" {l+x'Y
dx fi {^ + «)'^^
'' xf-+^-r
\a xl
x^dx
^' (a' - a;')''"
xdx zdz
8,
(a - xf '
x'dx
W x" I
9'
a' + (S - xY '"• (a + zf + (a - zY
(y — b)dy {z — c)dz
■ iy-by + {y + bY a^-az + z^'
(x - a)dx (y + a) dy
'3- ^{x - b) ■ '4. «»_(y+-6)5-
z'dz z'-dz
224 THM INTEaRAL CALCULUS.
136. Reduction of Rational Fractions in general. A ra-
tional fraction is a fraction whose numerator and denominator
are entire functions of the variable. The general form is
2'o + ?.« + ?,»" + •••+ ?»«" ~ D '
If the degree m of the numerator exceeds the degree n of
the denominator, we may divide the numerator by the de-
nominator until we have a remainder of less degree than n.
Then, if we put Q for the entire part of the quotient, and R
for the remainder, the fraction will be reduced to
D ^^ D
If we have to integrate this expression, then, since Q is an
entire function of z, the differential Qdx can be integrated
by § 119, leaving only the proper fraction -jr. Now, such a
fraction always admits of being divided into the sum of a
series of partial fractions with constant numerators, provided
that we can find th^oots of the equation D = 0. The theory
of this process belongs to Algebra, but we shall show by ex-
amples how to execute it in the three principal cases which
may arise.
Case I. The roots of the equation D = all real and un-
equal. Let these roots be a, ft, y ... 6. Then, as shown
in Algebra, we shall have
D^{x- a){x - /3){x -y). . . (x-e).
We then assume
D x-a^ x-^^ x-y^ " ' '
A, B, C, etc., being undetermined coeflBcients. To deter-
mine theni we reduce the fractions in the second member to
the common denominator D, equate the sum of the numera-
tors of the new fractions to R, and then equate the co-
efficients of like powers of x.
INTEOtBATION BY RATIONAL TBAmFOBMATIONS. 225
As an example, let us take the fraction
a; + 3
dx.
X — X
We readily find, by solving the equation x' — x = 0,
x' — X =: x(x — l){x -\- 1).
Assume
a: + 3 A
B
C
x' — X X X — 1 X -\-l
_ {A + B + C) x' + {B - 0) X - A
~ x' — X
Equating the coefficients of powers of x, we have ff Qp /
A+B+C=
0;
B-C =
1;
A = -
3;
whence B
= 2 and C
= 1. Hence
x + 3
x'-x'
_ 3 2
1
1
~ X ' x-1
1
x + l
and then,
by §130,
J)
//
^j>f- y ^,
a^-^.
•u>c
'r-%±^dx = - 3/^ + 2/^ + T-^
'J X' —X d X J X — \ tJ X-\-\
= - 3 log a: + 2 log (a; - 1) + log (2; + 1)+ log (7
G{x + V){x-\y
x'
log-
Integrate:
{x — Vjdx
'• x' -x-Q'
xdx
X" + X + 1
5" x'+x' -6x'
{x- + 2x*)dx
^" x' + 2x''-Sx'
x'dx
^' k' — (a + b)x + ab'
EXERCISES.
xdx
4-
6.
8.
(a; + x')dx
ix-l){x+l){x-2)ix+2y
x'dx
{x* ■\-x^)dx
x{x - l)(x + l)(x - 2)"
dx
x^ — (a-\- h^x^ + ahx'
226 TEE INTEGRAL CALCULUS.
Case II. Some of the roots equal to each other. Let the
factor X — a appear in D to the wth power. Then, if we
followed the process of Case I., we should find ourselves with
more equations than unknown quantities, because the n
fractions
X — a X — ax — a
would coalesce into one. To avoid this we write the assumed
series of fractions in the form
(a; - «)»^ (a; - «)»-' ^ • • • + a; - a + a: -yS + ^^^^
and then we proceed to reduce to a common denominator as
before. The coefficients A, B, etc., are now equal in num-
ber to the terms of the equation /) = 0, so that we shall have
exactly conditions enough to determine them.
As an example, let it be required to integrate
a;' - 5 ,
X — X — X -\-l
We have a;' - a;' - a; + 1 = (a: - 1)' (a; + 1).
We then assume
x'-5 A . -5 , C
(a; - 1)" (a; + 1) (a; - 1)' ' a; - 1 ' a;+ 1
_ (B + C)x' + (.4 - 20)x -i-A-B+O
(a;-ir(a; + l)
We find, by equating and solving,
A = -2;
B^+2;
0=-l.
Hence
a;' -5 -^ _j_ ^ 1
(a; - l)"(a; + 1) (« - 1)' ' a; - 1 x-\-l'
INTEaBATIOHr BT RATIONAL TRANSFOBMATIONB. 327
The required integral is
-^A-')-*+^/j#i-/jf-i
= ^4l^ + 2 log (x - 1) - log(a; + 1) + log C
2 , , Cix-iy
'log- —
X —1 ° X -\-l
EXERCISES.
Integrate:
dx dx
x{x+iy ■ x\x-iy
x'dx dx
^' {x - \)\x + %Y ^' {x - ay{x - by
(a + x)dx (a — x)dx
^' x\x - af x\x + ay{;x - h)'
Case III. Bnaginary roots. Were the preceding methods
applied without change to the case when the equation i) =
has imaginary roots, we should have a result in an imaginary
form, though actually the integral is real. We therefore
modify the process as follows:
It is shown in Algebra that imaginary roots enter an equa-
tion in pairs, so that if x ^ a -\- (Si (where i = V — 1) is a
root, then x = a — fti will be another root. To these roots
correspond the product
{x- a- ^i){x - a + ^i) = (a; - a)' + /?».
By thus combining the imaginary factors the function D
will be divided into factors all of which are real, but some
of which, in the case of imaginary roots, will be of the second
degree.
The assumed fraction corresponding to a pair of imaginary
roots we place in the form
A-[- Bx
{x-ay+^'
228 TBE INTEORAJu CALCULUS.
and then proceed to determine A and B as before by equa-
tions of condition. We then divide the numerator A -\- Bx
into the two parts
A + Ba and B{x — a),
the sum of which is ^ + Bx. Thus we have to integrate
/• A + Ba _, p B( x - a)dx , .
J (x- ay + z?'**^ + y (a - of + /J'- ^"^
The first term of (a) is, by methods ah-eady developed,
A -\- Ba , , ,.x — a
_^_tan<->-^,
and the second is
iB log {{x- ay + ^').
We therefore have, for the complete integral,
/
A ■\- Bx A-\-Ba, , ,.x—a
— tan'-'>-
BXERCISES.
I. / — ; :T-dx. 2. / -i -.
.»/ x — 1 "/a; — 1
The real factors of the denominator in Ex. 1 are (a' + l)(x + l)(x — 1).
We resolve the given fraction in the form
A + Bx C D
a' + l ^"a! + l'*'a!-l'
all
and find it equal to , . , + ^7^ H 5 • Then the integral is found
to be i log («!» + 1) + log (a!» — 1).
The factors of the denominator in Ex. 3 are «— 1 and 3?-\-x-\-\ =
r dx p (a:' + \)dx
^' J x'-\-l' ■ ^' J x' -'ix + 4"
Note that a; + 3 is a factor of the denominator in (4).
INTEGRATION BY RATIONAL TRANSFORMATIONS. 229
13 7. Integration ly Parts. Let u and v be any two
functions of x. We haye
d(uv) dv , du
az dx dx
By transposing and integrating we have
fufjlz = uv-fv^£dx + h, (1)
which is a general formula of the widest application, and
should be thoroughly memorized by the student. It shows
us that whenever the differential function to be integrated
can be divided into two factors, one of which f;^-^^;] can be
integrated by itself, the problem can be reduced to the inte-
gration of some new expression \v-^dx\.
The formula may be written and memorized in the simpler
form
/ udv = uv — I vdu, (2)
it being understood that the expressions dv and du mean dif-
ferentials with respect to the independent variable, whatever
that may be.
It does not follow that the new expression will be any easier
to integrate than the original one; and when it is not, the
method of integrating by parts will not lead us to the integral.
The cases in which it is applicable can only be found by trial.
The general rule embodied in the formulae (1) and (2) is
this :
Express the given differential as the product of one function
into the differential of a second function.
Then its integral will he the product of these two functions,
minus the integral of the second function into the differential
of the first.
230 TBE INTEGRAL CALCULUS.
EXAMPLES AND EXERCISES IN INTECRATION BY PARTS.
1. To integrate x cos xdx.
We have cos xdx = d- &ia x.
Therefore in (3) we have
u = x; v = emx;
and the formula hecomes
/ X cos xdx = I xd'smx = a; sin a; — / sin xdx
= a; sin a; + cos a; + h,
which is the required expression, as we may readily prove by
differentiation.
Show in the same way that —
2. I x sin xdx = — a; cos a; + sin a; + A.
3. I X sec' xdx = X tan x — (what ?).
4. I X sin X cos xdx = — \x cos 2a; + i sin 2a; + h.
5. /log xdx = a; log a; — / xd-logx= a; log a; — a; + 7i.
6. The process in question may be applied any number of
times in succession; For example,
/ x' cos xdx = I a;'^-sina; = x' Bin x — 2 I x sin xdx.
Then, by integrating the last term by parts, which we have
already done,
fx' cos xdx = x' sin a; + 2a; cos a; — 3 sin a; + h.
7. In the same way,
fx' cos xdx = / x'd- sin a; = x' sin x— 3 / x' sin xdx;
fx' sin xdx = — / x'd- cos a; = — a;' cos a; -j- 3 / a; cos xdx.
INTEGRATION BT RATIONAL TRANSFORMATIONS. 231
Then, by substitution,
/ x' cos xdx = {x' — 6a;) sin x + (3a;' — 6) cos x-\-'h.
8, In general,
I x" cos xdx = /a;"rf-sina;=a;"sina; — n /a;"-' sin xdx;
— / a;""' sin xdx = I a;"~'<Z-cos a;
= a;"-' cos X — {n — 1) I x"-" cos a;da;;
— Jx"-' cos a;rfa; =— a; "-'sin a; +(» — 2) / a;"-' sin a;<?a;;
/a;"-' sin a;rfa; =— a;""' cos a; +(w — 3) / a;"""* coaxdx.
etc. etc. etc.
Then, by successive substitution, we find, for the required
integral,
{a;"— w(w— l)a;"-«+M(«-l)(>i— a)(M— 3)a;"-*— . . .} sin a;
+ j«a;"-^ — m(w — 1) (w — 2)a;"~' + • • ■ } cos a;.
9. In the same way, show that
/ x" sin xdx =
{—a;"4-«(»*—l)^"~*— >»(»*— l)(w—2)(M—3)a:"~*-f. ..( cos a;
+ jwa;"-' — n{n — l)ln — 2)a;"~' +. • •} sin x.
/I Z' a;** "''^
a;" log xdx = — -— - / log xd' (a;" + *) = log x
fl -p It/ w — p X
1 /'a;"** a;» + i , a;"+'
7 / dx = — — :; log X — 7 — , ■ ,, .
n + lj X n + 1 ^ (» + 1)
II. rxe-'^dx=- r^xd-{e-'") = - ^^+i- A-^rfa;.
e~'^dx = .
Jlence jxe~^dx = —
xe'
232 THE INTEGRAL CALCULUS.
12. To mtegrate Q^6~'"dx when m is a positive integer, we
proceed in the same way, and repeat the process until we re-
duce the exponent of x to unity. Thus,
x'^e-'^dx = — :-- h - / x'^-^e-'^dx.
a aj
Treating this last integral in the same way, and repeating
the process, the integral becomes
a;'»e~'" mx'^-'-e-'^ m(m — l)x'^-''e-'"
a =^ ^ etc.
a a a
-ax
^-^i(a"2:"+OTa"— ■a;"'-'+»w(OT-l)a'»-''a;»-'+ . . .+ml).
a
13. From the result of Ex. 5 show that
y"(log xydx = x{r -21 + 2) +h,
where we put, for brevity, I = log x.
14. Show that, in general, if we put
w, =y (log a;)" dx,
then M„ = a;Z" — WM„_i;
and therefore, by successive substitution,
M„ = xil' — nl"-^ + n{n — 1)1"-' - . . . ± nl) + h.
15. Deduce {m + 1) fpx^dx = Fx'" + ^ - J'x'^ + ^dP.
16. Show that if P Pdx = Q,
then / Px^'dx = ga;" — n I Qx'^-^dx.
Also, if we have
/ Qdx - R; I Rdx = S, etc.,
then
fPa^dx = Q^ — nRx''-'^ + n{n — l)^"-^ — etc.
INTEGRATION OF IRRATIONAL DIFFERENTIALS. 233
CHAPTER IV.
INTEGRATION OF IRRATIONAL ALGEBRAIC
DIFFERENTIALS.
138. When Fractional Powers of the Independent Vari-
able enter into the Expression. In this case we may render
tie expression rational by reducing the exponents to their
least common denominator, and equating the variable to a new
variable raised to the power represented by this denominator.
Example. If we have to integrate
1 + ^^.
— ■ — -.dx,
1 + a;*
then, the L, 0. D. of the denominators of the exponents being
6, we substitute for x the new variable z determined by the
equation
x = z',
which gives dx = 6z^dz.
The differential expression now reduces to
,z' + z\
z' + l
By division this reduces to
dz.
6{Z' -Z' + Z' + Z'-Z- l)dz + ^j-p-^ + -5-p^.
Integrating and replacing z by its equivalent, .r*, we find
n+^dx = u - u + u + u - u - ex^
i/l_[_a;* 7 5 4 '3 2
+ 3 log (.T* + 1) + 6 tan<-« x^ + h.
234 THE INTEGRAL OALOULUa.
If the fractional exponent belongs to a function of x of the
first degree, that is, of the form ax + I, we apply the same
method by substituting the new variable for the proper root
of this function.
Example. To integrate
{a-\-hx)^dx
1 + (a + bx)'
We put {a + bx)^ = z; a + bx = z';
, 2zdz
ax = —J—.
The expression to be integrated now becomes
2z'dz _ 2( _ dz \
b{l + z')~b[ z'+lJ'
of which the integral is
~{z- tan(-'>2 + A) = J- j {a + bx)*-twi^-»{a + bx)*+h \ .
EXERCISES.
Integrate:
x^dx x^dx 1 — a;*
I. r— I — . 2. !• 3- \dx.
{a - x)'dx Jfl-xfdx_ g l + «-^,to.
'*■ 1 + a - «• ^- 1 _ (« _ xf ' (a - xy
7. i^^dx. 8. ^^^.. 9. J?£Z1_«M
xo. l±i^^dz. „.'illi£±f)-V
1 + (2 - c)* 1 + (a; + a)'
y a; , a;' -
12. — aa;. 13. ^dx.
X*
14.
]_
Vx ~ z*
(x - a)* - (a: - g)* ^^
(a; - a)* + (a; - a)*
INTEGRATION OF IRRATIONAL DIFFERENTIALS. 235
139. Gases when the Given Differential contains an
Irrational Quantity of the Form
Va-\-bx-{- ex'.
It is a fundamental theorem of the Integral Calculus that
if we put B = any quadratic function of x, then every ex-
pression of the form
F{x, VR)dx,
{F(x, VH) being a rational function of x and i/R), admits of
integration in terms of algebraic, logarithmic, trigonometric
or circular functions. But if R contains terms of the third
or any higher order in x, then the integral can, in general, be
expressed only in terms of certain higher transcendent func-
tions know as elliptic and Abelian functions.
We have three cases of a quadratic function of x.
First Case : c positive. If c is positive, we may render the
expression rational by substituting for x the variable z, de-
termined by the equation
Va -\- bx -\- ex' = Vc{x -\- z);
.' . a ■{• ix -{- ex' = ex" -\- 2cxz -\- cz'.
mi,- ■ cz' — a
This gives X = j^Tg- ; («)
a — bz + cz'j ,,.
Va + ix + cx' ^ - Va^^^. (.)
By substituting the values given by (a), (5) and (c) for the
radical, x, and dx, the expression to be integrated will become
rational.
Second Case ; a positive and c negative. If the term in x'
is negative while a is positive, we put
Va-^bz — ex' = Va-\- xz.
We thus derive a; =: — ;— | ; (a)
236 THE JNTEQBAL CALCULUS.
2( Vaz' -Vac- hz), ,,,
dx = -5^ ^,^^, dz; (b)
The substitution of these expressions will render the equa-
tion to be integrated rational.
Third Case : a and c both negative. If the extreme terms
of the trinomial are both negative, we find the roots of the
quadratic equation
— a -\- bx ~- ex' = 0,
which roots we call a and /?. We then have
— a-\-bx — ex' = e{a — x) (x — /3),
and we introduce the new variable z by the condition
V— a-\-bx — ex' = Ve{a — x) (x — /3) — Ve{x — a)z,
which gives z = ^. ' ^ ; (a)
»^- {z' + iy ' ^"^
^ ^a + bx-ex' ^ ^£^>; (e)
z +1 ^ '
substitutions which will render the equation rational,
140. We have already integrated one expression of the
dx
form just considered, namely, —^==^^= without ration-
Va -\-bx-\- ex'
alization. There is yet another expression which admits of
being integrated by a very simple transformation, namely,
de^- —^'—^ .
r Var' -\-br — 1
This is the polar equation of the orbit of a planet around
the sun. To integrate it directly, we put
INTEOBATION OF IRRATIONAL DIFFERENTIA '^^^,
1 , dz
z = —; dr = -..
We thus reduce the expression to
— dx
f:
Va-\-hz — x"
Proceeding as in §133, Case I., we find the value of the
integral to be
/:
dr , ,. %x — h , ,, 2 — br
= cos*" ' — — = cos^~ ' ^^^
r Var' + br-l V4:a + b'' r Via + b'
Thus, d-7r = cos(-» -^^JL=,
r VU + V
n being an arbitrary constant. Hence
= cos {a — Tt).
r Via + V
Solving with respect to r, we have, for the polar equation
of the required curve,
3 . .
^ ~ 5 + 4/(4a ■+ V) cos ((9 - n)' ^"^
This can be readily shown to represent an ellipse. The
polar equation of the ellipse is, when the major axis is taken
as the base-line and the focus as the pole,
*" ~ 1 4- e cos ^ ~ 2 , 2e '
~R 5\ + "7i T\ COS
a(l — 6 ) a(l — e )
Comparing with (a), we have
o
a(l — e°) = 7- = parameter of ellipse =^;
or e = -^-^ — 7 = eccentricity of ellipse.
•^38 THE INTEGRAL CALCULUS.
Irrational Binomial Forms.
141. General Theory, An irrational binomial differen-
tial is one in the form
(a + ix'^yz'^dx, (1)
in which m and n are integers positive or negative, while p is
fractional.
To find how and when such a form may be reduced to a
rational one, let the fraction p, reduced to its lowest terms, be
r
— ; and let us put
y=(a + bx'^y. (2)
This will give, when raised to the rth power and multi-
plied by x^dx,
{a + bx^yx'^dx = x-y'tiz, (3)
We readily find, from (2),
ix" = y' — a; (a)
dx =
hnx'
x^'fdx = 7-a;'"-"+*y'" + »~Wy;
or, substituting for x its value from (a),
a^y^dx = ^(^-^Y^^y^ + '-'dy. (4)
This last differential will be rational if is an in^
n
teger, which wiU be the case if ^ is an integer. We shall
call this Case I.
To find another case when the integral may be rationalized,
let us transform the given differential (1) by dividing the bi-
nomial by a;" and multiplying the factor outside of it by x"",
which will leave its value unchanged. It will then be
mTEQBATION OF IRRATIONAL DIFFERENTIALS. 239
{i-\-ax- ""Yx " + ''"dx, (1')
■vrhich is another differential of the same form in which n is
changed into — n and m into m + np. Hence, by Case I.,
this form can be made rational whenever — is an
n
integer; that is, when — — — \-pia such.
We have, therefore, two cases of integrability, namely:
Case I. : when — ^i^ = an integer.
n
Case II. : when — — — |-^ = an integer.
Eemaek. It will be seen that all differentials of the form
r
(a + Ix^Yx'^dx must belong to one of these classes, because
— '^^— is an integer when m is odd, and — ^^ H o ^^ sach
when m is even. In this statement we assume r to be odd,
because if it is even the original expression is rational.
143. If, in Case I., the integer is + 1, that is, if wi + 1
= n, then the expression can be integrated immediately.
For (4) then becomes
the integral of which, after replacing y by its value in (3), be-
comes
(« + hxrx^-^dx = ^j^f^^ + c. (5)
Again, if the integer in Case II. is — 1, we have
m -\- 1 -\- np = — n,
or JM + wp = — M — 1.
The expression (1) reduced to the form (1') will then be
(5 + aa: -")''« -»-'<?a; = - (5 + aa;-")" — d{l) -\- ax-").
240 TEE INTEGRAL CALOULUS.
which is immediately integrablej and gives by simple reduc-
tions
143. Forms of Reduction of Irrational Binomials. Al-
though the integrable forms can be integrated by the substi-
tution (2), it will, in most cases, be more convenient to ap-
ply a system of transformations by which the integrals can be
reduced to one of the forms just considered. The objects of
these transformations are:
I. To replace mhy m -\- n or vi — n;
II. To replace ;? by ;? + 1 or ^ — 1.
144. Firstly, to replace m by wi + n. Let us write, for
brevity,
X= a + Ja;",
which will give dX = Inx " ~ ^dx,
and the given differential will be
X'x'^dx,
which, again, is equal to
^.m — n + l /j.ln — n + 1
X''dX= ~' , , .■ <^(X^+')-
hn bn{p-\- 1)
Integrating by parts, we have
Since
X^ + ' = X^{a + W) = aX" + iXPx",
the last integral in the above equation is the same as
a f X^x '"-''dx + b C X'x'^dx,
of which the second integral is the same as the original one.
Making this substitution in (a), and then solving the equa-
INTEOBAriON OF IRRATIONAL DIFFERENTIALS. 241
tion so as to obtain the Talue of / X^x^dx, we find
/ X'x'^dx = T7 ; — ^— r^ — TT •^^^^ / ^'^a;''^«'a;. {A)
'J o(np-\- m -\-l) o{np-\-m-\-l)t/
Thus the given integral is made to depend upon another in
which the exponent of x is changed from m to m — n.
By reversing the equation we make the given integral
depend on one in which the exponent is increased by n. To
do this we change m into m -{• n all through the equation
(^4), thus getting
I j,Px'«+''dx=^, j 1 i^^-TT — \ , , -,^ / ^'a;'"^?^;-
«/ o{np-^m-}-n+l) o(np-{-m-\-n->rl)<y
Solving with respect to the last integral, we find
fx'-x-dx = 4^; - Hm+E±-+}) fx^x'^^^dx. (B)
The repeated application of [A) and {B) enables us to
make the value of the given integral depend upon other in-
tegrals of the same form, in which
»i is replaced by m-{-n; m-\-'^n; etc.;
or by m — n; m — %n; etc.
145. Next, to obtain forms in which p is increased or
diminished by unity, we express the given differential in the
form
X'-x'^dx = X'd
m + l
\m -)- 1/
Integrating by parts and substituting for dX its value
})nx''~^ dx, we have
fx^x-^dx = ^f^' _ JUlS- fx^-H-^^-dx. IV)
Now, we have
„m + » _ „n.„» _ a:"(X- «) _ Xa;" aa;"*
242 THE INTECHIAL GALGULUB.
and therefore, by multiplying by X^-^dz,
Substituting this value in (S), and solying as before with
respect to / X'afdx, we shall find
J wp + m + 1 Mp + m + 1^/ ' X '
in which p is diminished by unity.
If we write ^ 4-1 forjo in this equation, the last integral
will become the given one. Doing this, and then solving
with respect to the last integral, we find
fx^x-dx = - ?--p^, + "^+;+"+^ rX^^x-dx.(D)
J an{p-\-l) an{p-\-l) J ^ '
By the repeated application of the formula (C) or (2>) we
change
p mkop — 1, ^ — 3, ^ — 3, etc.,
or p into p -\-l, p -{■%, p -\-Z, etc.
146. To see the efEect of these transformations, let us
put, in the criteria of Cases I. and II., § 141:
I. — ■ — = I, an integer.
n
11. — ■ \-p = i', an integer.
Then when we apply formula {A) or (5), since we replace
m by »j — M or wi -}- re, we have, for the new integers:
n
^ mT^ + 1 ., ^^
n ^
It is also clear that by (C) and {D) we change II. by unity.
Thus, every time we apply formulae {A), (B), (C) or (D)
we change one or both of these integers by unity, so that we
may bring them to the values unity treated in § 142.
INTEGRATION OF IRRATIONAL DIFFERENTIALS. 243
147. Case of Failure in this Bedudion. If, in an integral
of Case II., i' is positiye, we cannot change it from zero to — 1
by the formula {A) or (C), hecanse, when — — 1- ^ = 0,
we have
m -\- 1 -\- np — 0,
and the denominators in {A) and (C) then vanish. In this
case we have to apply the substitution of § 141, without try-
ing to reduce the integral farther.
EXAMPLES AND EXERCISES.
1. To integrate
(a' ± x'fdz.
We see that if we diminish the exponent -J by unity, we
shall reduce the integral to a known elementary form of § 132.
So we apply (C), putting
m = 0; n = 2; p = i', a = a^; S = ± 1.
Then (C) becomes
J(,a±x)dx- g +2y(a.±^y
We therefore have, from § 132,
y (a' + x'fdx = \ I x{a' + x'f + «' log ^{x+{a^+ x'f) [ ;
/*(«' - x')^dx = ~ j x{a' - x'Y + a' sin^-" |- + A I .
Deduce the following equations:
2. / (c' — x') xdx =h — i{c' — a;')'.
3.f{o' + xrxdx =h+(^+^fl\
4. /{C + x'Yx-^dx = h + (^y^'.
_dx ^ ^^ _ ( c' + x')*
x'ic' + a;")* ~ ' " o^x
244 THE INTBQBAL CALCULUa.
7. y (a' - x')^dz = ia;(a' - x'f + K Bin <-« -.
Here apply formula ((7); in the following (4).
8. fil - x^fx'dx ^ h - (J + A)(i _ ^.)».
9. To reduce and integrate (1 + x')^x'dx.
Here m = S; n = 2; p = i; m -\- 1 = i = 2n. "We can therefore
reduce the form to Case I. by a transformation of m into m — n, for
which we may use either (a) or (A) of § 144. Using (a), we have
y (1 + a')* a^dx = (^+^')"°' ' _ ly (1 ^ j,j)8 a!;^.
The last integral can he immediately found, and gives for the required
integral
4(1 + »')«(!!» -A(l +»')'■ (o)
Using {A), we should find
y(l + «»)* !,^dx = (1 + »')« g - ^g), (J)
a form to which (a) can be immediately reduced.
The student will remark that the form (a) is reduced to (A) because
in the former the exponent of X is increased by 1, which often makes
the integration inconvenient. But when this increase of p does not in-
terfere with the integration, we may use (as) more easily than {A).
10. To reduce and integrate (1 + x^)*x^dx.
Applying {A), we find
y (1 + x^f afdx = <1 + ^'^ _ *y (1 + a,5)* aJijte.
A second application repeats the form (J) above, thus giving
y(i+.')V<*.==(i+.^)^g-S+j«^).
11. Reduce and integrate (1 + x')*x'^dx, where m is any
positive odd integer, and show that
INTEOBATION OP IBBATIONAL FUNOTIONB. 345
y (1 + a;')^ dx
■ j/a;"'-' (m-l)a:"'-' , (m-l)(OT-3)a;'^ \
^ ^ ' \m+2 (m + %)m ' {m+2)m{m-2)
Bemabk. Where the student is ■writing a series of transformations he
will find it convenient to put single symbols for the integral expressions
which repeat themselves. Thus:
rx''afdx = {l); /'x''a!'"-"(&=(a); etc.
Thus the equations of reduction in the present example may be written
^^'- m + 3 -^^+2^''^^'
XVrJ_»^3
m m
etc. etc.
12. Deduce the result
5ffl°
138
■j a;(a' + a;')* + a' log C(a: + V^T+x^) f .
246 TEE INTEGRAL CALCULUS.
CHAPTER V.
INTEGRATION OF TRANSCENDENT FUNCTIONS.
When the given differential contains trigonometric or other
transcendent functions of the variable more complex than the
simple forms treated in Chapter II., no general method of
redaction can be applied. Each case must therefore be
studied for itself.
148. To find the integrals
/ e"" cos nxdx and / e™" sin nxdx. (1)
Since we have
the integration by parts of these two expressions gives
/e"" cos wa; , M /> ^^ . ,
e"" cos nxdx = \-- e"^ sin nxdx;
/e"™ sin wa; n f ,^ ,
e"" sm nxdx = / e"" cos nxdx.
m mJ
Solving these equations with respect to the two integrals
which they contain, we find
/e'^im cos nx + n sin nx)
e""* cos nxdx — — ^^ ■ -'
m' + n'
/„, , , e""(»» sm nx — n eos nx)
e"" sin nxdx = — ^ 5-1 — ~ — ^J
(3)
which are the required values,
Eemaek. These integrals can also be obtained by substi-
tuting for the sine and cosine their expressions in termg <jf
imaginary exponentials, namely,
INTEGBATION OF TBANBCENDENT FUNCTIONS. 247
2 cos «« = 6"** + e ~"^,
a sin nx = T-(e"^ — e""^),
and then integrating according to the method of § 134. The
student should thus deduce the form (2) as an exercise.
149. Integration of sin" x cqs" xdx.
This form is readily reducible to that of a binomial, and
that in two ways. Since we have
cos xdx — d'sia. x,
cos a; = (1 — sin' a;)*,
we see that the integral may be written in the form
/
n- 1
(1 — sin' a;) a wa!^ xd'saxx;
or, putting y = sin x,
f{l-y')^y'-dy. (3)
By putting « = cos a; we should have, in the same way,
/ m — 1
(1 - z') ~»~z''dz, (4)
which is still of the same form, and is always integrable by
the methods already developed in Chapter IV.
If either m or w is a positive odd integer, then by develop-
ing the binomial in (3) or (4) by the binomial theorem we
shall reduce the expression to a series containing only posi-
tive or negative powers of x, which is easily integrable.
We can also, in any case, transform the integral so as to in-
crease or diminish either of the exponents m and n by steps
of two units at a time, as follows:
sin"" X cos" xdx = cos " ~ ' a; sin" xd' sin x
„ , ^in^+'a:
= cos"~'a!a , 1^.
w-f 1
Then, integrating by parts, we have
248 TEE IN2EQBAL CALCULUS.
A —T /sin"" + ^ a; cos""' a;rfa;. (5)
m -\-lJ ^ '
I sin" X cos" xdx
_ cos"~^ X sin "■ + ':!;
Because sin "" + ^2; = sin" x{l — cos" x), the last term is
equivalent to
'I — I/*-™ -«• w — !/>.„ „,
— r-^- / sm" 2; COS""* a; -^ I sm" a; cos" xdx.
m + It/ J» + It/
The last of these factors is the original integral. Trans-
posing the term containing it, we find
(ot + n) I sin" a; cos" xdx — sin" + 'a; cos"~*a;
+ (w — 1) / sin" a; cos ""' xdx, (6)
in which the exponent of cos x is diminished by 3.
We may in a similar way place the given difEerential in the
form
• ™ 1 7 cos " + ^ a;
— sm""^ xd -— — ,
« + 1
and then, proceeding as before, we shall find
(m + n) I sin" a; cos" xdx = — sin"~'a; cos" + ' x
-\- {m — 1) I siw^-' X cos" xdx, (7)
thus diminishing the exponent of sin x by 3.
By reversing these two equations we get forms in which
the exponents are increased by 2. "Writing n-\-2 f or m in
the first, and ?k + 2 for m in the second, we find
{n + 1) / sin" X cos" xdx = — sin" + ' a; cos" + ^ x
+ {m + n + 2)/' sin" a; cos " + ^ xdx; (8)
(m + 1) / sin" X cos" xdx = sin"*^ a; cos" + ^ x
-\- {m+n + 2) J sin "' + ^ x cos" xdx, (9)
INTEGBATION OF TRANSCENDENT FUNCTIONS. 249
150. Special cases of I sin" x cos" xdx.
If m is zero and n is positive, we derive, from (6),
/„ , sin a; COS""' a; , n — 1 r> „ , ,
cos" xdx — / cos"~* xdx;
n ' 11 J
/' » a J sin a; cos"-" » , w — 3 /» „ , -, \^^)
/cos"-^xdx = -s ^ / cos"-* xdx;
etc. etc. etc.
The integral to be found will thus become that of cos xdx
when n is odd, and that of dx, or x itself, when n is even.
The given integral is then found by successive substitution.
We find in the same way, frctm (7),
/. „ , cos a; sin "-^ a; , m — 1 /• . „ „ ,
sm™ xdx = / sin™ "xdx;
m m J
/■ ™_5 7 cosa;sin"^a; , ?w — 3 /> . „^ ,
sm ""-"»& = -^ s / sin"^a;rfa;;
etc. etc. etc.
From (8) and (9) we derive similar forms applicable to the
case of negative exponents.
(11)
EXERCISES.
I. / sin' X cos' xdx.
Ans. \ cos' X — ^ cos' x
2. 1 sin' X cos' xdx.
Ans. \ sin' x — \ sin'a;.
Poos' xdx
•^V sin* X '
. 3 sin' x — 1
Ans. „ ■ . .
3 sm' X
4. y sin' X tan' xdx.
/"cos' X -,
5. / V — ^—dx.
^ J tan' X
6. / e'" sin ^ydy.
7. re'' + '' cos (x + h)dx.
8. / e'" sin y cos ydy.
g.J'e-^coa' {y -\- a) dy.
JO. Derive the formulae of reduction
tan"*"'
Aan™ xd9^ = ' — xi~ "~ AaiJ""^^ ^^^\
II
12
250 TBE INTEGBAL CALCULUS.
and hence
/ tan" xdz = r / tan"~^ xdx.
These equatious may be obtained independently by putting tan" x
= tan " - 2 a!(sec' x—1); or they may be derived from (5).
Hence derive the integrals:
/ tan' xdx = ^ tan' x — log c sec x. (Cf. § 127)
. / tan* xdx = ^ tan' a; — tan x -\- x -\- h,
13. For all odd positive integral values of n,
/•, „ , tan'-'a; tan"-'^ , , ,
/ tan" xdx = s- + ... ± log c sec x.
J M — 1 n — 3
14. When 71 is positive, integral and an even number,
/», „ , tan"-'a; tan"~'a; , , ,
/ tan" xdx = = — h • • • ± tan x ± x.
J n — \ M — 3
15. When the exponent is integral, odd and negative,
/', „ , cot"~'a; , cot"-'a; , ,
/ iwx'^'xdx = z ^ . . . ± logcsma;.
J n — 1 n — 3
16. When the exponent is integral, even and negative,
/>, „ , cot"-^a; , cot"-'a; , , ^
/ tan-" xdx = — = ... ± cot a; q: a;.
J n — 1 n — 3
/■ » 7 cos xf . , , 4 . , , 4-3\
sm xdx = ---I sin a; + o sm ^ + oT^ )•
i8. I Bin' xdx
coBxf . , , 5 . , , 5-3 . \ , 5-32;
= ^[sinx + ^smx + -^^smx)+^:^^.
19. / sin" X cos" xdx = -- / sin" 2xdx
cos 2z sin"~' 2a;,w — l/>._„„,
— s^m h -5s — / sm"- " 2xdx.
3"+' n 3"w J
INTEOBATION OP TRANSCENDENT FUNCTIONS. 251
151. To integrate , . . ; — = ^— = du.
'' m sm X + n cos x
Dividing both terms of the fraction by cos" x, noticing that
= d'tan X and writing t = tan x, we find
cos X
dt
" = / — 573— i — 5- (12)
J mt -{-n ^
The integral is known to be (§ 138)
J^tan<-«^^,
mil n
so that we have
u = f , . , "^r , — = — tan'-" - tan z + h, (13)
J VI sm X -\- n cos x mn n > ' \ /
or tan x — — tan mn(u — h).
m ^ '
153. Inteq ration of — --r=^ .
rt + cos «/
We reduce this form to the preceding one by the following
trigonometric substitution:
a = «(cos" ly -j- sin" |?/);
h cos «/ = S(cos" ^y — sin" |y);
by which the expression reduces to the form
J {a - I) sin" iz/ + (a + h) cos" i^y' ^ '
which is that just integrated, when we put
a; = ly;
m = Va — i;
n = Va-\-i.
We therefore have
f^^ = ^_tan<-V^tan^ + A. (15)
J a-\-bQosy ^a' — b' a+ i '^ ' ^ ^
252 THE INTEGRAL CALCULUS.
153. If, in the form of §151, w" and «' have opposite
signs, or if in § 152 we have b > a, imaginary quantities will
enter into the integrals, although the latter are real. If, in
the first form, the denominator is m' sin' x — n' cos" x, we
shall have, instead of (12), the integral
J m't' - n' ~ ¥nj mf^n ~ %nj W+n ^§ ^^^^
mt -\- n
■M^^-:+''-
f
2mn ° mi — n
Hence, corresponding to (13), we have the result
dx _ 1 m tan ^ + n .
m' sin" x — n' cos' x ~ 2mn ° m tan x — n ^ '
If, now, in § 152, 5 > «, we write (14) in the form
_ 3 /• ^iy_
J {b — a) sin' i«/ — (a + b) cos' ^y'
and instead of (15) we have the result
/
dy ^/f I ^ log ^'^ ^ *^^^ ^■^+ '^'^ + 1 (17)
a+b cos y "*" V^,'_fl' ^ Vi-atan ^y- Vb -\- a
154. Infrgration of sin mx cos nxdx.
Every form of this kind is readily integrated by substitut-
ing for the products of sines and cosines their expressions in
sines and cosines of the sums and differences of the angles.
We have, by Trigonometry,
sin mx cos nx = ^ sin {m -{- n)x -\- ^ sin (in — n)x.
Hence
, cos (w/ + n)x cos (m — n)x , ,
Sm mx cos nxdx = -r- ; r ;t7^ T T ^
2(m -\- n) 2{m — n)
We find in the same way
sin (m. 4- 7i)x , sin (m — n)x , ,
cos mx cos nxdx = —rr^ ; r tt? — - — r — h "
2(to + n) 2(m — n)
sin (m -4- n)x , sin (tii — n)x , ,
sm mx sm nxax = ^ — , — r- H — ^7^ 7- + A,
2{m 4- V) 2{m — n)
f
f
INTEGBATION OF TBAN80ENDBNT FUNCTIONS. 253
155. Integration ly Development in Series. When the
given derived function can be developed in a convergent
series, we may find its integral by integrating each term of
the series. Of course the integral will then be in the form
of a series. The development of many known functions may
thus be obtained.
EXAMPLES AND EXERCISES.
I. We may find / sin xdx as follows: We know that
•/
X' , X° X'
a;" a;' x'
sin aiflla; = A + ^ - ^ + g| - etc..
which we recognize as the development of — cos x with an
arbitrary constant h -\-\ added to it.
Of course we may find / cos xdx in the same way.
dx
2. To integrate
1 + a;"
(1 + a;)-' = 1 - a; + a;' -a:' +
■/
1+a;
Now, we know that Jytt — ^°S (1 + *)• Hence (a) is
the development of log (1 -f x), when we put % = log 1 = 0.
The series (a) is divergent when a; > 1. In this case we
may form the development by the binomial theorem in de-
scending powers of x, thus:
ix + l)-^ = x-^ — x-' + x-^-x-^+ . . . .
Hence we derive, when a; > 1,
log(. + l) = log. + l-^4 + ^l^_l^+....
254 THE INTEaBAL CALCULUS.
The arbitrary constant is zero because, when x is infinite,
log (a; + 1) — log X is infinitesimal.
%. To find / ' = sin <~" x in a series.
Hence
r dx . , ,, , 1 a;V 1-3 a;', 1-3-5 a;' ,
The arbitrary constant is zero by the condition sin<~" = 0.
This series could be used for computing w by putting x =
i, because i = sin 30° = sin -. But its convergence would
be much slower than that of some other series which give the
value of Tt.
dx
rive the expansion
4. From the equation / — = log {x-^- Vl + x") de-
t/ Vl-\-x'
1 x" , 1-3 x' 1-3-5 a;'
log ^^+vr+^)=x--.^ + ^-^.^ - ^.^.^ + . .
5. By expanding , = £Z-tan<~" a;, derive
tan<-» x = x-ix' + ix'-^x''+ . ...
Derive:
/• '^^ -z,r I a^° 1-3 a:' 1-3-5 x"
• J l^r+^'~ 2-5 +2-4-9 2-4-6 13+*
7. fe-'''dx = h + x-^+^,-^^,+ ....
DEFINITE INTEGRALS. 255
CHAPTER VI.
OF DEFINITE INTEGRALS.
156. In the Differential Calculus the increment of a
variable has heen defined as the difference between two values
of that variable. Let us then suppose m to represent any
variable quantity whatever, and let us suppose u to pass
through the series of values
"oJ ^1) ^J> M„ . . • Un.
Then we shall have
Jw„ = w, - w„;
^M, = Mj — M^;
^«, = Ms - M.;
Taking the sum of all these equations, we have
Au„ + Zu^ + All, + . . . + ^«<„_i = «„ - w,;
That is, the difference between the two extreme values of a
variable is equal to the sum of all the successive increments
by which it passes from one of these values to the other.
The same proposition may be shown graphically by sup-
posing the variable to represent the distance from the left-
hand end of a line to any point upon the line. The differ-
ence between the lengths Au, and Au^ is evidently Au^
-\-Au,-\-...+ Au,.
, I Am, I AMi I Am, I Au, I An, | ....
_j «„ Ui u, «s ttj «5
Since the proposition is true how small soever the incre-
ments, it remains true when they are infinitesimal.
256
TEE INTEGRAL CALCULUa.
157. Differential of an Area. Let P^PP' be any curve
whatsTer, and let us inyestigate the differential of the area
swept over by the ordinate y
XP. Let us suppose the
foot of the ordinate to
start from the position X„,
and move to the position
X. During this motion
XP sweeps over the area
X^P^PX, the magnitude
of which will depend upon
the distance OX, and will ^'°- ^''•
therefore be a function of x, which represents this distance.
Let us put
u = the area swept over;
y = the ordinate XP.
Then, if we assign to x the increment XX', the corre-
sponding increment of the area will be XPF'X'. Let us call
y' the new ordinate X'P'. It is evident that we may always
take the increment XX = Ax ao small that the area XPP'X'
shall be greater than yAx and less than y' Ax or vice versa.
That is, if y' > y, as in the figure, we shall have
yAx < Au < y'Ax.
Now, when Ax approaches the limit zero, y' will approach
y as its limit, so that the two estremes of this inequality yAx
and y'Az will approach equality. Hence, at the limit,
du = ydx. (1)
That is, the area u is such a function of x that its differen-
tial is ydx, and its derivative toith respect to x is y.
From this it follows by integration that
: / ydx + h
(2)
is a general expression for the value of the area from any
initial ordinate, as X^P^ to the terminal ordinate XP.
DEFINITE INTBQBAL8.
257
158. The Conception of a Definite Integral. Suppose
the area X„P„PX=m
to be divided up into
elementary areas, as in
the figure. This area
will then be made up of
the sum of the areas of
all the elementary rect-
angles, plus that of the o^
triangles at the top of s'"'- «•
the several rectangles. That is, using the notation of § 156,
we have
« = y,Ax, + y^Ax^ + y,Ax^ + • • • + y„_,^a;„_, + T,
T being the sum of the areas of the triangles; or, using the
notation of sums,
« = n — 1
u= 2 y^Axi + T.
io
Now, let each of the increments Axt become infinitesimal.
Then each of the small triangles which make up T will be-
come an infinitesimal of the second order, and their sum T
will become an infinitesimal of the first order. We may
therefore write, for the area u,
x=OX x=OX
u = lim. 2 yAx = 2 ydz.
x=OXo x=OXo
That is, u is the limit of the sum of all the infinitesimal
products ydx, as the foot of the ordinate XP moves from X„
to X by infinitesimal steps each equal to dx.
Such a sum of an infinite number of infinitesimal products
is called a definite integral.
The extreme values of the independent variable x, namely,
6>X„ Ea;„ and 0X= a;,, are called the limits of integration.
The infinitesimal increments ydx, whose sum makes up the
definite integral, are called its elements.
17
258 TEE INTEGBAL CALCULITS.
159. Fundamental Theokem. The definite integral
of a continuous function is equal to the difference between the
values of the indefinite integral corresponding to the limits of
integration.
To show this let us write (f>{z) for y, and let us put, for the
indefinite integral,
'*(f)(x)dz = F(z) + c.
/'
Now, as already shown, this is a general expression for the
area swept over by the ordinate y = (p(x), when counted from
any arbitrary point determined by the constant c. If we
count the area from X„P„, the area will be zero when x = x„;
that is, we must have
F{x,) +c = 0,
which gives c = — F{x„).
If we call a;, the value of x at X, we shall have
u = Area X,P„PX = F{x,) + c = F{x,) - F{xX (3)
which was to be proved.
We therefore have a double conception of a definite in-
tegral, namely:
(1) As a sum of infinitesimal products;
(2) As the difference between two values of an indefinite
integral;
and it will be noticed that the identity of these two concep-
tions rests on the theorem just enunciated.
Notation. The definite integral is expressed in the same
form as the indefinite integral, except that the limits of inte-
gration are inserted after the sign / above and below the line;
thus,
r(f>{x)dx
means the integral of cp{x)dx taken between the limits x„ and
x^, the first being the initial and the second the terminal limit.
DEFimTE INTEaSALS. 259
Example of the Identity of the Two Conceptions of a Defi-
nite Integral. The double conception of a definite integral
just reached is of fundamental importance, and may be
further illustrated analytically. To take the simplest possible
case, consider the definite integral
Jx~
adx,
a being a constant. By definition this means the sum of all
the products
adx -\- adx -\- adx + . . . ,
as X increases from «„ to x^. The sum of all the doi^a must
be equal to a;, — x^ (§ 156), Hence
a{dx -\- dx -{■ dx -\- dx -\- . . . ) = a{x^ — a;„).
But we have for the indefinite integral
/
adx ■■
and the definite integral is therefore, by the theorem,
ax^ — ax, or a{x^ — a;„),
as before.
160. Differentiation of a Definite Integral with respect to
its Limits. — Because the definite integral / ydx = u means
the sum of all the products ydx as x increases by infinitesimal
increments from the lower limit x, to the upper limit a;,, or
u = y^dx + y'dx + y"dx + . . . + y^'^^dx,
therefore, assigning an increment dx^ to the terminal limit
x^ will add the infinitesimal increment y^dx^ to u (see Pig. 48).
That is, we shall have
du = y,dx„ or ^ = y, = 0(a;j. (4)
In the same way, increasing the initial limit x, by dx, will
take away from the sum the infinitesimal product y,dx,, so
260 THB INTEGRAL CALCULUS.
that we shall have
|=-,.= -0K). (5)
The equations (4) and (5) give us the derivatives of the
definite integral
u= I <p{x) . dx
with respect to its limits x^ and a;„.
161. Examples and Exercises in finding Definite Inte-
grals.
The fundamental theorem gives the following rule for form-
ing definite integrals:
1. Form the indefinite integral.
2. Substitute for the variable with respect to which we inte-
grate, firstly, the upper limit of integration; secondly, the
lower limit.
3. Subtract the second result from the first. The difference
will be the required definite integral.
1. / x'dx = ix^' — ix,\
2. j xdx = 1(5' — a"). 3. / xdx = i.
4. / sin xdx = — cos n -\- cos = 2.
5. / cos xdx = sin ^Tt. 6. / azdz = ia{a* — S').
' sin 2xdx. 8. / cos 2xdx.
t/45°
sin' xdx. 10. / cos' xdx,
«/o
II. / X Bin xdx. 12. / z COB zdz.
t/o t/o
13. I z' sin zdz. 14. / 2' cos «<?«.
DEFINITE INTEGBALS. 261
15. / z' cos 2zdz. 16. /* z' sin %zdz.
t/O t/o
/^dx „ />" ,
— . 18. / nz^dz.
/^ dx p^' dz
—, ,• 20. / -5 -.
21. / cosa;<?a;, 22. / siuxdx.
25. / (a; — «)«». 26. / ydy.
t/a — b t/a — X
^1 — a; _a + c
27. / (is — l)'(?a;. 28. / [x — a){x — c)dz
31. r^'smaxdx. 32. f^ cos {a + x)dx.
COS (x + ?/)</a; = sin 2y.
V
34. Show that r f{x)dx = — C f{x)dx.
35. Deduce / c~''dy = l.
/■" 1
36. Deduce / e~'"'c?w = -.
t/o a
37. Deduce / e^dy = 1.
t/ — QO
38. Deduce / e- ^''ydy = ^.
39. Deduce f -— — ^ = tt.
t/_oo J- + 2
«^ ^2 ;r
40. Deduce /
41, Deduce / - = n.
^ J- a Va' - z'
•+" dz
262
THE INTEGRAL CALCULUS.
163. Failure of the Metliod when the Function becomes
Infinite. It is to be noted that the equivalence of the two
conceptions of a definite integral does not necessarily hold
true unless the function ;/ or (p{x) is continuous and finite
between the limits of integration. As an example of the
failure of this condition,
consider the function
y
{x - af
the curve representing
which is shown in the
margin.
The indefinite integral is
u
- 1 ydx = c —
X
a
MA
Fig. 49.
To find the value of this integral between two such limits
as and k, h being any quantity OM less than a, we put
a; — and x = h, and take the difference as usual. Thus
1 1^ k
a
u ■■
(5)
a — k a a{a — k)'
!Now, if we suppose k to approach a as its limit, so that
a —k shall become infinitesimal, then the area tc will increase
without limit, as we readily see from the figure as well as by
the formula.
But suppose h > a; for example, k = 2a. Then the
theorem would give
,2a
1 _1 __3
/o ' a a a
a negative finite quantity; whereas, in reality, the area is an
infinite quantity.
The theorem fails because, when x ^= a, y becomes infinite,
so that ydx is not then necessarily an infinitesimal, as is pre-
supposed in the demonstration.
DEFINITE INTEGRALS. 263
163. Change of Variable in Definite Integrals. When,
in order to integrate an expression, we introduce a new vari-
able, we must assign to the limits of integration the values of
the new variable which correspond to the limiting values of
the old one. Some examples will make this clear.
Ex. 1. Let the definite integral be
dx
£
/o a-\-x
Proceeding in the usual way, we find the indefinite integral
to be log (a + x), whence we conclude
£^^=^^%^'^-^^%'^^^^^'^-
But suppose that we transformed the integral by putting
y'B.a-{-x; dy — dx.
Since, at the lower limit, a; = 0, we must then have y — a for
this limit, and when, at the upper limit, a; = a, ve have
y — 2a. Hence the transformed integral is
rdy
'y'
which we find to have the same value, log 2.
Ex. 2. u — I ^ sin x{l — cos x)dx.
We may write the indefinite integral in the form
/ sin xdx -\- I cos xd(cos x).
In the first term x is still the independent variable. But,
as the second is written, cos x is the independent variable.
Now, for
a; = 0, cos x = 1;
and for x = -^, cos x = 0.
Hence, writing y for cos x, the value of u is
u= 1^ sin xdx -\- j ydy = 1 ~ ^ = ^.
264 THE INTEGRAL GALGULU8.
Bekabe. The variable with respect to which the integra-
tion is performed always disappears from the definite integral,
which is a function of the limits of integration, and of any
quantities which may enter into the difEerential expression.
Hence we may change the symbol of the variable at pleasure
without changing the integral. Thus whatever be the form
of the function <p, or the original meaning of the symbols x
and y, we shall always have
jT <p{x)dz =J^ 4>{y)dy =£ "^{y + a)dy, etc.
164. Subdivision of a Definite Integral. The following
definitions come into use here:
1. An even function of a; is a function whose value remains
unchanged when x changes its sign.
2. An odd function of x is one which retains the same
absolute value with the opposite sign when x changes its sign.
As examples: cos x is an even, sin x an odd, function.
Any function of x* is even; the product of any even func-
tion into X is odd.
It is evident, from the nature and formation of a definite
integral, that if we have a sum of such integrals,
r (p(x)dx 4- /* <t>{x)dx + / (f>{x)dx +... + / <l){x)dx,
J a «/6 Jo Jg
in which the upper limit of each integral is the lower limit
of that next following, this sum is equal to
<t){x)dx.
f
This theorem may often be applied to simplify the expres-
sion of the integral in cases where the values of 0(a;) repeat
themselves.
Theoeem I. If cj){x) is an even function of x, then, what'
ever he a,
<p{x)dx = 2 / <p{x)dx.
DEFINITE INTEGRALS. 265
Because 0(— a;) = ^(a;), it follows that for every negative
yalue of x between — a and the element of 0(a;)c?a; will be the
same as for the corresponding positive value of x. Hence the
infinitesimal sums which make up the value of / cf){x)dx will
be equal to those which make up / ^{x)dx. Therefore
/a fji nd nf^
^{x)dx = / 4>{x)dx + / <p{x)dx = 3 / 4>{^)dx,
Theorem II. If(l>{x) is an odd function of x, then, what-
ever be a,
/ + »
(p(x)dx = 0.
a
For in this case each element (p{— x)dx will be the negative
of the element (j){x)dx, and thus the positive and negative
elements will cancel each other.
EXERCISES.
Show that / e'^'x^dx = I (log — j dz.
Substitute X = log -.
2. Show that whatever be the function 0, we have
j (j}(sm. z)dz = / <p{cos xdx).
As an example of this theorem.
'*"« + 5 cos" x^^ _ pi^a -\- I sin" x
b sin" X
n^^a -t- COS" a; , _ pt^a +
Jo a — b COS" X ~Ja a —
dx.
The truth of this theorem may be seen by showing that to each ele-
ment of the one integral corresponds an equal element of the other.
Draw two quadrants; draw a sine in one and an equal cosine in the other.
Any function of the sine is equal to the corresponding function of the
cosine. We may fill one quadrant up with sines and the other with
cosines equal to those sines, and then the two integrals will be made up
of equal elements.
266 THE INTEOBAL CALCULUS.
To express this proof analytically, we replace s by a new variable y
— \ic — X, which gives sin x = cos y; dx = — dy; and then we invert
the limits of the transformed integral, and change y into x in accordance
with the remark of the last article.
^ IT
3. Show that I /(sin x)dx = 2 i^fism x)dx.
4. Show that / 0(sin x) cos xdx — 0.
5. Show that if be an odd function, then
/ 0(cos x)dx = 0.
6. Show that the product of two like functions, odd or
even, is an even function, and that the product of an even
and an odd function is an odd function.
7. Show that when is an odd function, 0(0) = 0.
165. Definite Integrals through Integration ly Parts, —
In the formula for integration by parts, namely,
/ udv = uv — vdu,
let us apply the rule for finding the definite integral. To ex-
press the result, let us put
(uv)^ and (««)„, the values of uv for the upper and lower
limits of integration, respectivelyj
I udv and / vdu, the values of the two indefinite in-
tegrals for the upper limit, a;,;
fudv and / vdu, the values of the integrals for the lower
limit, x„.
We then have, by the rule of § 161,
Jf udv = I udv — I udv
= (m«), — / vdu — {uv), + / ^du
= (uv\ — {uv), — I vdu.
t/x«
DEFINITE INTEQBALS. 267
In order to assimilate the form of this expression to that of
a definite integral, it is common to write
{uv)\ = (uv), - {uv\.
EXAMPLES AND EXERCISES.
1. We have found the indefinite integral
/ log xdx = X log X — I dx.
If we take this integral between the limits a; = and x = l,
the term x log x will vanish at both limits, so that
{x log a;), - {x log a;)„ = 0.
Hence j log xdx — — j dx=— l-\-0=—\.
2. To find the definite integral,
I sin" xdx.
In the equation (11), § 150, the first term of the second
member vanishes at both the limits x — and x = 7t. Hence
/ sai!^xdx=. / sin^^^aj^^a;.
Jo m Jo
Writing »i — 2 f or m, and repeating the process, we have
m — Z
r 5bi'"~^xdx = X i sva.'^~^xdx',
Jo m — 2 Jo
I sm!^~^xdx — 7 / sin™~'a;(?a;:
Jo m — 4:Jo
etc. etc.
If m is even, we shall at length reach the form
I dx = 7t — = Tt.
Then, hj successive substitution^ we shall hav§
268 THE INTEGRAL CALCULUS.
Ja m(m — 2)(»i — 4) ... 3
If m is odd, the last integral mil be j sin xdz = + 2, and
we shall hare
r "sin™ xdx = s ("^ - ^)(^» - 3) ... 2
i/o m(m — 3)(m — 4) ... 3
3. From the equation (6) of § 149 we have, by forming the
definite integral and dividing hj m-\-n,
/»" /sin"'+'a;cos"~*a;\''
/ sin" z cos" xdx = I ;
Ja \ m-\-n Iq
A "^ — / sin" X COS""' xdx.
m + nJo
Since sin ?r = sin = 0, the first term of the second
member vanishes between the limits, and we have
/*"" n — 1 p""
I sin" a; cos" a!<?a; = ; — / sin" a; cos"~*a;£?a;.
Jo m + nJo
Writing n — 2, and then m — 4, etc., in place of n, this
formula becomes
f sin" a; cos""' a;<fa; = — ; ^/ sin"a; cos""* a;t?a;:
Jo m + n — 2 Jo
I sin" X cos""* xdx = — ; r / sin" x cos""' xdx;
Jo m + n — 4: Jo
etc. etc.
If n is odd, the successive applications of this substitution
will at length lead us to the form
XT 1
sin" X cos xdx = — r^ (sin"+' tt — sin" + ' 0) — 0;
and thus, by successive substitution, we shall find all the in-
tegrals to be zero,
DEFINITE INTEGRALS. 269
If n is even, we shall be led to the form
I sin" xdx,
which we have just integrated. Then, by successive substi-
tution, we find
/"
sm" X cos" as
{n - 1)(to - 3) . ■ ■ 1
{m + n){m + w — 2) . . , (wi + 2),
dx
'i)Jo
sin" xdx.
4. To find /7-5— i — iw-
Jo («' + «)
We transform the differential thus:
/dx _ 1 />(a;' + a" — a;') <?a;
1^+7^" ~ d'-J («' + d'Y
Integrating the last term by parts, we have
r x'dx _^ r _Jixdx__ _l r d- jx'+d')
J (a;' + ay ~ %J ^(a» + a;")» ~ 2 «/ * (a;" + a")"
=\Sxd.
(x'+ff")"-^ 2(w-l)"(a;" + «'')"-'
1 — w
1 r dx_
abstituting this value of the
Substituting this value of the last term in (a), we have
dx _ \ X
(a;' +^)» ~ %a\n - 1) (a;" + «')"-»
^ a'V 2(w - V)]J {x^ + «")»-!•
Passing now to the limits, we see that the first term of the
second member vanishes both for a; = and for a; = oo . We
also have
1 2«-3
1-
2(w - 1) 2(« - !)•
270 THE INTB&RAL OALGULTTS.
Hence we have the formula of reduction
n-^__dx__ _ 2w — 3 /»" dx
Jo {x'-\-a'Y~%{n-l)a'Jo (a' + a;"p^i" ^''>
We can thus diminish the exponent by successive steps
until it reaches 2. The formula {b) will then give
dx n
X (a;" + ay ~ 2^'Jq ^
+ x'~ 4a'"
Then, by successive substitution in the form (b), we shall
have
P'° dx ^ {2n - 3)(2w - 5) ... 1 tt
Jo {x' + a')''~ (2n-2)(2n-i). . .2-2a"'-'- ^"^
If in (c) we suppose a = 1, and write the second member
in reverse order, we have
Jo (1
dx _ 1-3-5 ■ . . (2w-3) jr
+ x'y~ 2-4:-6 . . . (aw-a)"2"
Let us apply to the indefinite integral the formula (A),
§ 144. We have in this case
a = l; b = —1; n = 2; p = — i.
The formula then becomes
r x^dx _ _ a;"-' Vl — a;' m- 1 f x'^-Hx
In the same way
P x'^-^dx _ _ x^-Wl-x" TO-3 P x'^-^dx
J if\—x^ ~ 'f"— 2 TO — 3t/ 4/i":ir^'
Continuing the process, we shall reduce the exponent of x
to 1 if m is odd, or to if m is even. Then we shall have
Taking the several integrals between the limits and 1, we
•^-=i"^
(0)
DEFINITE INTEGBALS. 271
note that in (a) the first term of the second member vanishes
at both limits, while {b) gives
/>! xdx _ />! dx _ 1
Jo Vl-x'~ ' J Vl-x'~ ^^'
We thus have, by successive substitution,
_ p x'''+^dx _ 2n{2n - 8) (2w - 4) 2
y^ + '=Jo |/izr^ - (a«+l)(3w-l)(2w-3) ... 3'
a^t^g: _ (2w-l)(2re-3)(2M-5) . ■ . 1 5
yrir^» ~ 2n{2n — 2) (3m — 4) 2 '2
Let us now consider the limit toward which the ratio of
two values of y^ approaches as m increases to infinity. We
find, from (a),
.Vm _ m — l
Vm-, m '
a ratio of which unity is the limit.
2fext we find, by taking the quotient of the equations (c),
n _ {2-4- 6 . ■ . {2n — 2)-2nY y^
2 - ts^tt. . (2w - \)\\2n + 1)-^;;:^^'
Since, when n becomes infinite, the ratio y^^ '■ .Van + 1 ap-
proaches unity as its limit, we conclude that \Tt may be ex-
pressed in the form of an infinite product, thus:
Tt 4 4' 6' 8' 10" , . ^ .,
2 ^ 3 "3^5 -S^r 7^9 -901 ' ' ' <'^ '^^fi^'*^^-
This is a celebrated expression for n, known as Wallis's
formula. It cannot practically be used for computing n,
owing to the great number of factors which would have to
be included.
272 THE INTEGRAL GALCULUB.
CHAPTER VII.
SUCCESSIVE INTEGRATION.
166. Differentiation under the Sign of Integration. Let
us have an indefinite integral of the form
u = I (j){a, x)dx — F(a, x), (1)
a being any quantity whatever independent of x. It is evi-
dent that u will in general be a function of a. We have
now to find the differential of u with respect to a.
The differentiation of (1) gives
d'u _ d<t)(^a, x)
dadx da
_ d''u ^du ^du , , .,
Because -^—7- = ^a^— = -^» j~» ^^ ame, when we consider
-3— as a function of x (cf. § 51),
Jdu\d^_a^^ dcl>{a,x) ^^^
\da I dxda da
Then, by integrating with respect to x,
^ ^ p d<P{a,x)
da J da
in which the second member is the same as (1), except that
0(a, x) is replaced by its derivative with respect to a. Hence
we have the theorem:
The derivative of an integral with respect to any quantity
which enters into it is expressed hy differentiating with re-
spect to that quantity under the sign of integration.
SVCCESaiVE INTEasATION. 273
167. This theorem being proved for an indefinite inte-
gral, we have to inquire whether it can be applied to a definite
integral. If we take the integral (1) between the limits a;,
and a;,, and put m„ and m, for the corresponding values of u,
we have, for the definite integral,
r ^4>(oL, x)dx = F{a, a;,) - F(a, x,) = u,-u, = m,'.
Then, by differentiation,
<Zm/ _ dF{a, a;,) dF(a, a;„)
da da da
Comparing (1) and (3), we have
(3)
rd^{a^ ^^ ^ dF{a, a;) .
v da da '
da da
whence, if a;, and a;„ are not functions of a,
^^ d<p{a, x) ^^ ^ dF{a, a;, ) _ dFja, x,) ^
Jx„ da da da ' ^ '
Hence from (3) we have the general theorem
Da / (t>{a, x)dx = I Da.4>{a, x)dx.
That is, the symbols of differentiation and integration with
respect to two independent quantities may be interchanged in
a definite integral, provided that tJie limits of integration are
not functions of the quantity with respect to which we differ-
If the limits a;, and a;„ are functions of a, we have, for the
total derivative of m/ with respect to a (§ 41),
t?M„' _ !du^\ du^ dx^ du„ ' dx^
da ~ \ da ) dx^ da dx„ da'
By §160 we have
18
274 THE INTEQBAL CALCULUB.
Thus from (3) and (4) we have
«?M.' /^idip(a,x), , ,, .dx, ,, ,dx. ...
This formula is subject to the same restriction as the
theorem for the value of a definite integral; that is, (l>{a, x)
and its derivative with respect to a must be finite and con-
tinuous for all values of x letween the limits of integration.
If this condition is not fulfilled, (5) may fail.
EXERCISES.
Differentiate:
1. / — ; — with respect to a. Ans. — / 7 — ; — r-..
2. j{x + aydx with respect to a. Ans. n l{x-\-a)''-^dx.
3. Ma;' -l-a;y)'(?a; with respect toy. Ans. 2 r{x^-\-z''y)dx.
4. r x'dx with respect to a. Ans. a'.
5. / x'dx with respect to a. Ans. 8a'.
ajVa; with respect to or. Ans. =a"(2a" + ^—l).
And show that we have the same results in the first three
cases whether we integrate the differential with respect to a
or y, or differentiate the integral.
168. The preceding method enables us to find many
integrals, indefinite and definite, by differentiating known
integrals with respect to constants which enter into them.
Thus, by differentiating with respect to a the integral
fef^dx = -6-^+0,
8U00ESSI7M INTEGRATION. 275
we find, after adding the constants of integration,
/.'.«^..= g-^-f + !>- + .;
etc. etc.
wMcli leads to tlie same results as integration by parts, and
is shorter.
169. The following is an instructive application of this
and other principles. We shall hereafter show that
From this it is required to find the value of / e~" ** dy.
If we put
x = ay,
whence dy = — ,
the corresponding indefinite integral will be
f'-"''y=\f'
e *' dx.
N"ow, when y = ± oo , we have also a; = ± oo . Hence
r^\-'''y'dy=ir^\--dx=^.
tJ—CXi ^t/— CO d
By differentiating with respect to a, and simple reductions,
we find
and from this,
etc. etc.
2x
cos ax:
276 TSE mTEORAL CALCULUS.
KXERCISES.
1. By differentiating the integrals
y cos axdz = - sin ax,
fsia axdx = cos ax,
twice with respect to a, prove the formulae
/• , , fx' 2\ . , 2x
I X cos axdx =1 5 1 sm ax -\ — 5
r ■, ■ , (2 x'\ , 2x .
I X sm axdx =[—, cos ax + -5 sm ax.
J \a al a
Thence show that we have
Jy^ cos ydy = (y' - 2) sin y + 2y cos y;
Jy' sin ydy = {2 - y') cos y + 2y sin y.
2. Prove the formulae:
/o 1 />" 1
e'-^dx =-; (b) / x^'dx = - -,;
00 a fj — 00 a
K'e-^Ja; = -,; (rf) / x'^^dx = (- 1)"-^.
3. Show that the preceding formulae are true only when a
is positive, and find the following corresponding forms when
a has the negative sign:
/»°° , 1 />" ,1
/ e-'^dx = -; I xe-'^dx = -,;
i/o a i/o a
r x'e-'^dx = -,; C x'e-'^dx = ~r ; etc.
Jo a Jo a'
4. By differentiating the form of § 132, namely,
dx
/(a'
{a' - xy
with respect to a, show that
dx
sm
(-«:
/(«'
{a' - xy ~ a'(a' - x')*'
SUCCESSIVE INTEORATION. 277
170. Double Integrals. The preceding results may be
summed up and proved thus: Let us have an integral of the
form
u = J<p{x, y)dx, (1)
and let us consider the integral
J'udy or fy/'Pi^' y)d^j^y>
which, for brevity, is written without brackets, thus:
J J(p{x, y)dxdy.
This expression is called a double integral.
Theoeem. The value of an indefinite double integral re-
mains unchanged when we change the order of the integra-
tions, provided that we assign suitable values to the arbitrary
constants of integration.
Let us put
V = J<p(x, y)dy,
u retaining the value (1). The theorem asserts that
/ udy = I vdx.
Call these two quantities U and V, respectively. We then
have, by differentiation,
dU d^U du ^, ,
— =v ^^=— = ct>(x
dx ' dydx ~ dy ~ ^^ ' "/•
Therefore, because of the interchangeability of differentiations,
^dU ^dV
'dx _ 'dx
dy ~ dy '
Then, by integration with respect to y,
— - — 4-c-
dx ~ dx '
278 THE INTEGRAL CALCULUS.
and, by integration with respect to x,
U— V+cx + c'.
Putting c = and c' — 0, we have U= F, as was to be
proved.
mi. By the process of successive integration thus indi-
cated we obtain the value of a function of two variables when
its second derivative is given. The problem is, having an
equation of the form
^"^ <l>{x,y), (3)
dxdy
where 0(a;, y) is supposed to be given, to find u, as a func-
tion of x and y. This we do by integrating first with respect
to one of the variables, say x, which will give us the value
of T— , because the first member of (2) is D~r^. Then we in-
dy ' dy
tegrate with respect to y, and thus get «.
As an example, let us take the equation
d'u „ ^ du , -
xy , or d.-^ = xy dx.
dxdy '^ ' 'dy
Integrating with respect to x, we have
%=ry+'' (3)
h being a quantity independent of x, which we have common-
ly called an arbitrary constant. But, in accordance with a
principle already laid down (§118), this so-called constant
may be any quantity independent of x, and therefore any
function we please to take of y.
Next, integi'ating (3) with respect to y, and putting
Y=
= Jhdy,
we find u = Ix^y' + F-f X,
in which X is any quantity independent of y, and so may be
an arbitrary function of x. Moreover, since h is an entirely
arbitrary function of y, so is Y itself.
SUGOESSrVE INTE&BATION. 279
The student should now prove this equation by difEerenti-
ating with respect to x and y in succession.
1'73. Triple and Multiple Integrals. The principles just
developed may be extended to the case of integrals involving
three or more independent variables. The expression
cl){x, y, z)dxdydz
fff^
means the result obtained by integrating ^{x, y, z) with re-
spect to X, then that result with respect to y, and finally that
result with respect to z. The final result is called a triple
integral.
If we call F{x, y, z) the final integral to be obtained, we
have.
d^F{x, y, z) ., ,
and the problem is to find F{x, y, z) from this equation when
0(a;, y, z) is given.
Now, I say that to any integral obtained from this equation
we may add, as arbitrary constants, three quantities: the one
an arbitrary function of y and z; the second an arbitrary
function of z and x; the third an arbitrary function of x and y.
For, let us represent any three such functions by the symbols
[y, 2]> b, ^], [x, «/],
and let us find the third derivative of
F{x, y, z) + {y, z] + [«, x] + \x, y'] = u
with respect to x, y and z. Differentiating with respect to
X, y and z in succession, we obtain
du _^ dF(x, y, z) d[z, x] d[x, yl _
dx ~~ dx dx dx '
O'u _ d'Fjx, y, z) d'lx,y] ^
dxdy ~ dxdy dxdy '
d'u _ d'F{x, y, z) _
dxdydz dxdydz '
an equation from which the three arbitrary functions have
entirely disappeared.
280 THE INTEOBAL CALCULUS.
It is to be remarked that one or both of the variables may
disappear from any of these arbitrary functions without chang-
ing their character. The arbitrary function of y and z, being
any quantity whatever that does not contain x, may or may
not contain y or z, and so with the others.
As an examplCj let it be required to find
u = I I I {x — a){y — b){z — c)dxdydz.
Integrating with respect to z, and omitting the arbitrary
function, we have
J fW - "){y - *)(« - cfdxdy.
Then integrating with respect to y,
^=J-^{x-a){y-bY{z-oy;
which gives, by integrating with respect to x, and adding the
arbitrary functions,
?« = i(x - afiy - l)\z - cY + [y, z] + [z, x] + [x, y].
The same principle may be extended to integrals with re-
spect to any number of variables, or to multiple integrals.
The method may also be applied to the determination of a
function of a single variable when the derivative of the func-
tion of any order is given.
EXERCISBS.
3. / / I xy'z'dxdydz. 4. I j j-^dxdydz.
5. I I I {x — ay{y — b){z — cydxdydz.
6. ff{x~aydx\ 1. J J J(z-\-'hydz\
Ans. (6). ^^{x — ay -\- Ox -\- C, Cand C" being arbitrary
constants.
SUCCESSIVE INTEGRATION. 281
173. Definite Dotiile Integrals. Let U be any function
of X and y. By integration with respect to a*, supposing y
constant, we may form a definite integral
/
Udx= U'.
From what has been shown in § 163, Eem., U' will be a
function of y, x^ and x^. We may therefore form a second
definite integral by integrating U'dy between two limits «/„
and y^. Thus we find an expression
f U'dy =11 Udxdy,
which is a definite double integral.
The limits a;„ and x^ of the first integration may be con-
stants, or they may be functions of y.
If they are constants, the two integrations will be inter-
changeable, as shown for indefinite double integrals.
If they are functions of y they are not interchangeable, un-
less we make suitable changes in the limits.
174. Definite Triple and Multiple Integrals. A definite
integral of any order may be formed on the plan just described.
For example, in the definite triple integral
r / / (j)(x, y, z)dxdydz
the limits a;„ and a;, of the first integration may be functions
of y and z; while i/„ and y, maybe functions of z. But z^ and
^j will be constants.
So, in any multiple integral, the limits of the first integra-
tion may be constants, or they may be functions of any or all
the other variables. And each succeeding pair of limits may
be functions of the variable which still remain, but cannot
be functions of tho§e with respect to which we have already
integrated,
282 THE LNTEGBAL CALCULUS.
EXAMPLES AND EXERCISES.
I. Find the values of
/ / xy'dxdy and / / xy'dxdy.
It will be seen that in the first form the limits of x are
constants, and in the second, functions of y.
First integrating with respect to x, we have for the indefi-
nite integral
/ xy'dx — ^x'y'',
and for the two definite integrals
/ xy^dx = ^a'y^,
/ xy^dx = \y\
f/y
Then, integrating these two functions with respect to y,
we have
££\y-'dxdy = if'y^dy = ^b\
Let us now see the effect of reversing the order of the in-
tegrations. First integrating with respect to y, we have
^xy'dy = ixb' = U.
Then integrating with respect to x, we have
f Udx ^J^Tx^dydx = \a^h\
the same result as when we integrated in the reverse order
between the same constant limits.
2. Deduce f ^ f cos (a; + y)dxdy = — 3.
SUCCESSIVE INTEGRATION. 283
3. Deduce / / cos {x — y)dxdy = + 4.
4. Deduce / / {x — a){y — b)dxdy = ^a'b'.
5. Deduce f''J"'{x - a)(y - b)dxdy = \{2ab-a'){2ah-h').
6. Deduce / " / ^ {x—a){y—b)dxdy = a'b — iab'— fa'.
1*75. Product of Two Definite Integrals.
Theoebm. The product of the ttoo definite integrals
I ^Xdx atid I ' Ydy is equal to the double integral
I "' I 'XYdxdy, provided that neither integral contains
the variable of the other.
For, by hypothesis, the integral / Xdx = C7 does not con-
tain y. Therefore
U f Tdy^f UYdy = / / XYdxdy,
Jva vVo vy„ ux„
as was to be proved.
176. The Definite Integral I e~'' dx. This integral,
which we have already mentioned, is a fundamental one in
the method of least squares, and may be obtained by the ap-
plication of the preceding theorem. Let us put
k= r'^'°e-''''dx = % r^'^e-'^'dx^'H r'^"'e-^^dy.{%lU)
Then, by the theorem,
lc^= 4 r^\-^'dx r^\-''dy =4 r^" Z'+V^'+^'Wy.
i/o Jo t/o t/o
Let us now substitute for y a new variable t, determined
by the condition
y = tx.
284 THE INTEGRAL CALCULUS.
Since, in integrating with resiject to y, we suppose x con-
stant, we must now put
dy = xdt.
Also, since t is infinite when y is infinite, and zero when y is
zero, the limits of integration for t are also zero and infinity.
Thus we hare
k' = ir^'" r^''e-'^'+''^xdxdL
Since the limits are constants, the order of integration is
indifEerent. Let us then first integrate with respect to x.
Since
xdx = \d-x^ = 2(1^1') ^' (1 + *')'^'>
the integral with respect to x is
Then, integrating with respect to t.
Hence / e~'''dx = V7r.
£
BECTIFICATION OF OUBVES. 285
CHAPTER VIII.
RECTIFICATION AND QUADRATURE.
177. The Rectification of Curves. In the older geometry
to rectify a curve meant to find a straight line equal to it in
length. In modern geometry it means to
find an algebraic expression for any part of
its length.
Let us put s for the length of the curve
from an arbitrary fixed point C to a vari-
able point P- If P' be another position
of the variable point, we shall then have no. so.
As = PP'.
If PP' becomes infinitesimal, it has already been shown
(§ 79) that we have, in rectangular co-ordinates.
ds = Vdd' + dy^ = /l + i^£ldx = 1/1+ i^)dy, (1)
and, in polar co-ordinates,
If both co-ordinates, x and y, are expressed in terms of a
third variable u, we have
*■ = &»)■+ (I* )^
The length of any part of the curve is then expressed by
286
THE INTEGRAL CALCULUS.
the integral of any of these expressions taken between the
proper limits. Thus we have
or
•=/{S)+(f)T-
(3)
In order to effect the integration it is necessary that the
second members of (3) shall be so reduced as to contain no
other variable than that whose differential is written; that is,
we must have
ds=f{x)dx; f{y)dy; f{e)dd; or f{u)du.
Then we take for the limits of integration the values of
X, y, 6 or u, which correspond to the ends of the curve.
178. Rectification of the Parabola. From the equation
of the parabola
y' = 2px
we derive ydy = pdx.
We shall have the simplest integration by taking y as the
independent variable. We then have
rfs = 1 1 +(|)' I 'dy; pds = \p' + y']'dy.
The formula ((?) of § 145 gives
/(/ + y')'dy = iy{p' + y')* + Wf^l ^^
{a)
(y+y')'
,«\*"
The method of § 132 gives
dy
fr-rir^i = A - log {{p' + ff - y)
"^ {p +y)
= A - log ^ + log (( p' + y')* + y).
BECTIFICATION OF CUBVES. 287
Thus, putting h' = ip {h — log^), the indefinite integral
of (a) is
s = h' + i|(/ + f)' + ^p log ((y + ff + y).
The arbitrary constant h' must be so taken that s shall
vanish at the initial point of the parabolic arc. If we take
the vertex as this point, we must have s — Qloxy = 0. Then
h' - -ip log p.
We therefore have, for the length of a parabolic arc from
the vertex to the point whose ordinate is y,
s = \lif + ff + \plo,i^+f±JL. (4)
179. Rectification of the Ellipse. The formulae for rec-
tifying the ellipse take the simplest form when we express the
co-ordinates in terms of the eccentric angle u; then (Analyt.
Geom. )
a; = a cos m; y = b sin u.
We then have
dx = — a sin udu; dy = h cos udu.
Then if e is the eccentricity, so that a'e' = a' — V,
ds = (a' sin' u -\-b' cos' uydzi = «(1 — e' cos' u)*du;
s = a / {1 — e' cos' «)*
du.
This expression can be reduced to an elliptic integral: a
kind of function which belongs to a more advanced stage of
the calculus than that on which we are now engaged.
It may, however, be approximately integrated by develop-
ment in series. We have, by the binomial theorem,
(1 — e' cos' m)* = 1 — s-e' cos' u — ^r— ; e* cos* u
— a.A.a ^ COS U — etc.
288 THE INTEQBAL CALCULUS.
The terms in the second member may be separately in-
tegrated by the formulae (6), § 149, by putting m = and
n — 'H, i, 6, etc. We thus find
2 / 008° udu — sin u cos u-\-u;
4 / cos* udu = sin m(cos'' if + f cos u) -\- fw;
etc. etc. etc.
Since at one end of the major axis we have m = and at
the other end u = re, we find the length of one half of the
ellipse by integrating between the limits and n. Since
sin u vanishes at both limits, we have
j cos' udu = x-7r;
1"3
cos' udu = :r-^;r;
3-4
1-3-5
We thus find by substitution that the semi-circumference
of the ellipse may be developed in powers of the eccentricity
with the result
A 1 , 3 . y-b . \
s = a7r\l - -,e - ^^,e - ^^r:^e -. . .j.
180. T/ie Cycloid. The co-ordinates x and y of the cy-
cloid are expressed in terms of the angle u through which
tlie geiierating circle has moved by the equations (§80)
X =? «(«< — sin m);
y = a(l — cos m).
Hence
ds' = dx* + dy' = «'{(! - cos uf + sin' u\du'
— 3a' (1 — cos u)du,^ = 4a' sin' ^u.du''.
By extracting the root and integrating,
s = h — 4,a cos iu.
BBCTIFICATION OF 0URYB8. 289
If we measure the arc generated from the point where it
meets the axis of abscissas, that is, where w = 0, we must
have s = f or ?{ = 0. This gives
and s = 4a(l — cos ^m) = 8a sin" ^.
This gives, for the entire length of the arc generated by
one revolution of the generating circle,
s = 8a;
that is, four times the diameter of the generating circle.
181. The Archimedean Spiral. From the polar equation
of this spiral (§ 83) we find
dr = add.
Hence ds = a(l + d'fdd.
Then the indefinite integral is (§ 147, Ex. 1)
s = 1 1 «(i + n' + log o{d +(1 + e')f [ .
If we measure from the origin we must determine the value
of by the condition that when 6 = 0, then s = 0. This
gives log (7=0; . ■ . C = 1.
If instead of 6 we express the length in terms of r, the
radius vector of the terminal point of the arc, we shall have
s = ^-{a+r) +^logi -> .
183. The Logarithmic Spiral. The equation of this
spiral (§ 83) gives
^a = ale = Ir.
da
Hence ds = {l-\- Vfrdd.
To integrate this differential with respect to 6 we should
first substitute for r its value in terms of 6. But it wUl be
19
290 THE INTEGRAL CALCULUS.
better to adopt tlie inverse course, and express d6 in terms of
dr. We thus have
(1 + I'Y
ds = = dr',
(1 + l\i
whence s = ^ — —r + *»>
s„ being the value of s for the pole.
If we put y for the constant angle between the radius
vector and the tangent, then (§§ 90-92) ;=cot y, and we have
s = r sec 7 + *o'
Between any two points of the curve whose radii-vectors
are r„ and r, we have
s = (r, — r„) sec y.
Hence the length of an arc of the logarithmic spiral is pro-
portional to the difference iefween the radii-vectors of the ex-
tremities of the arc.
EXERCISE.
1. Show that the differential of the arc of the lemniscate is
add
Vl - 2 sin' e
(This expression can be integrated only by elliptic func-
tions.)
183. The Quadrature of Plane Figures. In geometrical
construction, to square a figure means to find a square equal
to it in area. The operation of squaring is called quadrature.
In analysis, quadrature means the formation of an algebraic
expression for the area of a surface.
In order to determine an area algebraically, the equation
of the curve which bounds it must be given. Moreover, in
order that the area may be completely determined by the
bounding line, the latter must be a closed curve.
Then whatever the form of this curve, every straight line
QUADRATURE OV PLANE MQURES.
291
s
/
T
/
/
o
v
—a
Xo
Xi
Fio. 51.
must intersect it an even number of times. The simplest
case is that in which a line paral-
lel to the axis of Y cuts the bound-
ary in two points. Then for
every value of x the equation of
the curve will give two values of y
corresponding to ordinates termi-
nating at P and Q. Let these
values be y„ and y^.
Then, the infinitesimal area in-
cluded between two ordinates infinitely near each other will
be
The area given by integrating this expression will be
in which the limits of integration are the extreme values of z
corresponding to the points X„ and X„ outside of which the
ordinate ceases to cut the curve.
The same principle may be applied by taking (a;, — x^dy
as the element of the area. We then have
o- = r\x, - x,)dy.
If the curve is referred to polar
co-ordinates, let S and T be two
neighboring points of the curve,
and let us put
r=OS;
r'=OT;
/I e = angle SOT.
If we draw a chord from S to T,
the area included between this chord
and the curve will be of the third
order (§ 78). The area of the triangle formed by this chord
Fig. !
292
THE INTEGRAL CALCULUS.
and the radii vectors will be ^rr' sin Ad. Now let /id be-
come infinitesimal. OS will then approach r as its limit;
the ratio of sin /J^ to Ad itself will approach unity, and the
area of the triangle will approach that of the sector. Thus
we shall have, for the differential of area,
da = ^"dd.
If the pole is within the area enclosed by the curve, the
total area will be found by integrating this expression be-
tween the limits 0° and 360°. Thus we have, for the total
area.
'dd.
184. The Parabola. As the parabola is not itself a closed
curve, it bounds no area. But we may find the area of any
segment cut off by a double ordinate
MH. The equation of the curve gives,
for the two values of y,
y, = + V2px;
Hence
y«
da = V^. x*dx.
The indefinite integral is
Fio. S3
a = iV2^+ C.
For the area from the vertex to JO^we put a;, = OX, and
take the integral between the limits and a;,. Calling this
area a^, we have
0", = ^ 1^3^.2;, = ^x^y^ = fa;, X2y,.
Because 2y, = il/iV, it follows that the area^^J/7V= 2a;,y,.
Hence:
Theokeit. The area of a parabolic segment is two thirds
that of its circumscribed rectangle.
QUADRATURE OF PLANE FIGURES. 293
185. The Circle and the Ellipse. Eef erring the circle
of radius a to the centre as the origin, the values of y will be
y=±{a'- cc')K
Hence
fiy,-y„)dx = 2f(a'-xydx
X
= x(a' - x')i + a' sin <-«- + h.
^ ' a
This expression, taken between appropriate limits, will
give the area of any portion of the circle contained between
two ordinates.
Taking the integral between the limits — a and + « gives,
for the area of the circle,
o- = a' sin<-« (+ 1) - a' sin'"" (- 1) = na\
The Ellipse. From the equation of the ellipse referred to
its centre and axes, namely,
El + ^-l
we find V = ± - Va' — a;'.
•^ a
The entire area will be
/ + o 5 /1 + a ,
• a Cio —a
The last integration is performed exactly as in the case of
the circle.
186. The Hyperbola. Since the hyperbola is not a closed
curve, it does not by itself enclose any area. But we may
consider any area enclosed by an hyperbola and straight lines.
Let us first consider the area APM contained between the
curve, the ordinate MP, and the segment AM ot the major
294
THE INTEGRAL CALCULUS.
axis. The equation of the hyperbola referred to its centre
and axes gives, for the value of
y in terms of x.
^ a
If we put a;, for the value of
the abscissa OM, then, since
OA — a, the area AMF will be
equal to the integral
- f^\x' - a')*dx;
Fio. 64.
and for the definite integral between the limits a and x,
Area APM-
1 te
3 a
(z' - a")
ah
"2
Now, ^xy is the area of the triangle 0PM; we therefore
conclude that the second term of the expression is the area
included between OA, OP and the hyperbolic arc AP.
Much simpler is the area included between the curve, one
asymptote, and two parallels
to the other asymptote. The
equation of the hyperbola re-
ferred to its asymptotes as
axes of co-ordinates (which
axes are oblique unless the
hyperbola is equilateral) may
be reduced to the form fio. 55.
xy =
ab
2 sin o^
QUADRATURE OF PLANE FIGURES. 295
a being the angle between the axes. We readily see that the
differential of the area is ydx X sin a instead of ydx simply.
Hence for the area we have
jy sin adz = j ^-dx = -^ log ex.
If we take the area between the limits OM=x„ and OM
= a;,, the result will be
/^i ab -, db ., X.
X Tx^^^^ii''^^:
We note that this area becomes infinite when a;„ becomes
zero or when a;, becomes infinite, showing that the entire area
is infinite.
187. The Lemniscate. The equation of this curve in
polar co-ordinates is (§ 81)
r° = a' cos 2ft
It will be noted that r becomes imaginary when 6 is con-
tained between 45° and 135°, or between 235° and 315°.
The integral expression for the area is
^fr'dd = ia'fcos 2edd = Ja' sin 2(9.
To find the area of the right-hand loop of the curve we
must take this integral between the limits = —i5° and 6 =
-h 45°, for which sin 2(9 = - 1 and -f 1. Hence
Half area = ^d';
Total area = a'.
Hence the area of each loop of the lemniscate is half the square
on the semi-axis.
188. The Cycloid. By differentiating the expression for
the abscissa of a point of the cycloid we have
dx = a{l — cos u)du.
Hence
296 THE INTEGRAL CALCULUS.
Cydx =a' /"(l— cos uydu=a'' y (|— 3 cos m + -J cos %u)du.
The indefinite integral is
|«{ — 3 sin M + i sin 2v.
To find the whole area we take the definite integral between
the limits and 2;r. Thus we find
Area of cycloid = 3;ra%
or three times the area of the generating circle.
EXERCISES.
I. Show that the theorem of § 184 is true only of the pa-
rabola.
To do this we must find what the equation of a curve must be in order
that the theorem may be true. The theorem is
J ydx-
Differentiating both members, we have
y^ = ixdy +
• -0 — ^
' ' y x'
Then, integrating both members,
log y'' = log ex ; .• . ^ = ex,
c being an arbitrary constant. This is the equation of a parabola whose
parameter is ic.
2. Show that the equation of a curve the ratio of whose
area to that of the circumscribed rectangle ia m : n must be
of the form
OUBATUBE OF VOLUMES.
397
CHAPTER IX.
THE CUBATURE OF VOLUMES.
189. General Fortnulce for Gubature. In the ancient
Geometry to cube a solid meant to find the edge of a cube
whose volume should be equal to that of the solid. In Ana-
lytic Geometry it means to find an expression for the volume
of a solid.
Let us have a solid the bounding surface of which is de-
fined by an equation between rectangular co-ordinates. Let
the solid be cut by a
plane PL parallel to the
plane of TZ, and let u
be the area of the plane
section thus formed. If
we now cut the solid by
a second plane, parallel
to PL and infinitely near
it, that portion of the
solid contained between
the planes will be a slice of area u and thickness dx, dx being
the infinitesimal distance between the planes.
If, then, we put v for the volume of that part of the solid
contained between any two planes parallel to YZ, we have
Tie. so.
and
dv = udx,
w = / udx,
(1)
a:„ and a;, being the distances of the cutting planes from the
prigin 0,
298
THE INTEGRAL CALCULUS.
If we take for a;„ and a;, the extreme values of x for any
part of the solid, the aboTe expression will give the total vol-
ume of the solid.
In order to integrate (1), we must express m as a function
of X. That is, we must find a general expression in terms of
X for the area of any section of the solid by a plane parallel
to that of XT. This is to be done by the equation of the
bounding surface of the solid.
Of course we may form the infinitesimal slices by planes
perpendicular to the axis of Y ot of Z as well as of X,
190. The Sphere. The equation of a sphere referred to
its centre as the origin is _ [Zj yy
a;' + y' + a' = a'.
If we cut the sphere by a plane
PMQ parallel to the plane of
YZ, and having the abscissa OM
= X, the equation of the circle of
intersection will be
y' + z' = a" — a?;
that is, the radius MP of the circle will be Va^ — x', and its
area will be 7t{a' — a;'). Hence the differential of the vol-
ume of the sphere will be
dv = n{a^ — x')dx,
and the indefinite integral will be
V = 7r{a'x — ix') +0.
The extreme limits of x for the sphere are
x„= — a and a;, = + «.
Taking the integral between these limits, we have
Volume of sphere = ^yra',
Fio. 57.
OVBATURE Of VOLUMES.
200
Let the pyramid be placed
191. Volume of Pyramid.
witli its vertex at the ori-
gin, and its base parallel to
the plane of XT. Let ub
also put h = OZ its alti«
tude; a, the area of its base.
Let it be cut by a plane
EFGH parallel to its base.
It is shown in Geometry
that the section EFGH is
similar to the base, and that
the ratio of any two homologous sides, as EFaudAS, is the
same as the ratio OL : OZ. Because the areas of polygons
are proportional to the squares of their homologous sides,
.-.Area EFGH : Area ABGD = OL' : OZ'.
Putting Area ABGD — a, OL = z and OZ = h,
Area. EFGH ^^.
Fig. 58.
The volume of the pyramid is therefore
That is, one third the altitude into the base.
The same formulae apply to the cone.
193. TJie Ellipsoid. The equation of the ellipsoid re-
ferred to its centre and axes is
a, h and c being the principal semi-axes.
If we cut the ellipsoid by the plane whose equation is
X = x', the equation of the section will be
b' '^ c'
1-
300
TBB INTEQUAL CALOULXfa.
This is the equation of an ellipse whose semi-axes are
■Vc?^'
and
V^
Hence its area is
ar5c(a' - x")
Fio. eg.
Then, by integration between the limits —a and +a, we find
Volume of ellipsoid = ^nabc.
From the known expression for the area of an ellipse (nab)
it is readily found that the volume of an elliptic cylinder cir-
cumscribing any ellipsoid is "iiTtabc. Hence we conclude:
The volume of an ellipsoid is two thirds that of any rigJit
elliptic cylinder circumscribed about it.
193. Volume of any Solid of Revolution. In erder that
a solid of revolution may have a well-defined volume it must
be generated by the revo-
lution of. a curve or un-
broken series of straight
or curve lines terminating
at two points, Q and R,
of the axis of revolution.
As an element of the volume we take two planes infinitely
near each other and perpendicular to the axis of revolution.
Every such plane cuts the solid in a circle. If we place the
origin at 0, take the axis of revolution as that of X, and let
OM = a; be the abscissa of any point P of the curve, and
MP = y its ordinate,
then the section of the solid through M will be a circle of ra-
dius y, whose area will therefore be Try'.
Hence the volume contained between two planes at distance
dx,wii.l be
Tty'dx,
and the volume between two sections whose abscissas are x„
and a;, will be
V = I mfdx.
(1)
OUBATUBE OW VOLUMES.
301
If the two co-ordinates are expressed in terms of a third
variable u by the equations
we have
X = 0(m), y = ip{u),
dx = 0'(m)c?m.
Putting M, and m, for the values of u corresponding to a;„
and »„ the expression (1) for the volume will become
V=7i r\ip{u)Y(t>'{u)du.
(2)
The equations (1) and (3) give the volume AA'B'B gen-
erated by the revolution
of any arc ^.B of the
given curve, and of the
ordinates MA and NB
of the extremities of the
arc. The limits of in-
tegration for X are OM
= x„ and OH = a;,. To
find the entire volume generated we must extend these limits
to the points (if any) at which the curve intersects the axis of
revolution.
194. TJie Paraboloid of Revolution. The equation of the
parabola being y' = 2px, we readily
find from (1) a result leading to the
following theorem, which the student
should prove for himself:
Theorem. The volume of a para-
boloid of revolution is one half that
of the circumscribed cylinder.
195. The Volume Generated by
the Revolution of a Cycloid around
its Base. From the equations of the cycloid in terms of
Fio. 61.
302 Tm: INTEGRAL CALCULUS.
the angle through which the generating circle has moved,
we find the element of the volume to be
<?F= 7ta'{i — cos uydu.
Hence
V = no? / (1 — 3 cos M + 3 cos' u — cos' u)du.
By the method of §§ 149, 150, with simple reductions, we
find
/ cos" udu = |m + J sin %u;
I cos' udu = / (1 — sin' u)d. sm m = sin m — ^ sin' u
= I sin M + -^ sin 3m.
We thus find, for the indefinite integral,
V = ;ra'(f M — J^- sin m + 1 sin 2m — -^ sin 3m).
The total volume formed by the revolution of one arc of
the cycloid is found by taking the integral between the limits
M = and M = 27r. The volume thus becomes
V=57t'a',
from which follows the theorem:
The volume generated hy the revolution of a cycloid around
its base is five eighths that of the circumscribed cylinder.
196. The Hyperboloid of Revolution of Two Nappes.
This figure is formed by the revolution of an hyperbola about
its transverse axis. The general expression for the volume is
found to be
F=|^;(a;'-3a'x + A),
h being the arbitrary constant of integration. If we consider
that part of the infinite solid cut ofE by a plane perpendicular
to the transverse axis, we must determine h by the condition
CUBATUBE OF VOLUMES. 303
that V shall vanish when x = a, because then the plane will
be a tangent at the vertex of the hyperboloid, and the volume
wiU become zero. This condition gives
n ■= 3a' -a' = ^a\
Thus we have
V - ^[(a:' - U'x + 3«') = g(a; - a)\x + 2a). (1)
By the same revolution whereby the hyperbola describes an
hyperboloid of revolution the asymptotes will describe a cone.
Let us compare the volume just found for the hyperboloid
with that of the asymptotic cone, cut off by the same plane
which cuts off the hyperboloid. The equation of the generat-
ing asymptote being
ay = hx,
we find for the volume of the cone
The difference between (1) and (2) will be the volume of
the cup-shaped solid formed by cutting the hyperboloid out
of the cone. Calling this volume V", we find
V" = 7th\x - f«). (3)
This is equal to the volume of a circular cylinder of which
the diameter is the conjugate axis of the hyperbola, and the
altitude x — fa.
This result is intimately associated with the following
theorem, the proof of which is quite easy:
If a plane perpendicular to the axis of revolution cut an
hyperbola of two nappes and its asymptotic cone, the area of
the plane contained between the circular sections is constant
and equal to the area of the circle whose diameter is the con-
jugate axis.
804
THE INTEGRAL CALCULUS.
4
p
f
o
M
X
Fia. 62.
197. Ring-shaped Solids of Revolution. If any com-
pletely bounded plane figure APQB revolve around an axis
OX lying in its own plane, but
wholly outside of it, it will describe
a ring-shaped solid.
To investigate such a solid, let the
ordinate MP cut the figure in the
points Q and P, and let us put
y, = MQ; y,^MP.
The points P and Q will describe two circles which will
contain between them the sectional area
Taking two ordinates at the infinitesimal distance dx, the
corresponding infinitesimal element of volume will be
dV=n{y:-y,')dx. (1)
The integral
V=7t r\y,' - y^)dx = n ["'{y^ + y^) (y, - y^^dx
will express the volume of that part of the solid contained be-
tween the two planes whose respective abscissas are a;„ and a;,.
By taking for x„ and x^ the abscissas of the extreme points
A and B, V wUl express the total volume of the solid.
198. Application to the Circular Ring. Let the figure
.45 be a circle of radius c, whose centre is at the distance b
from the axis of revolution. Let us also put
a = the abscissa of the centre.
We then have
y^
y. + y.
y, - y.
Vc'- (x - a)';
h + VT'-
2b;
{x - a)';
2Vc'-(x- a)';
CUBATUnE Of VOLUMMB. 305
F= 4;r5 /""[c' - {x - ayfdx.
tyXft
The limits of integration for the whole volume are
a;„ = a — c and x^ — a-\-c.
If we put
z = x — a,
the total volume will become
V=4.7iljJ''^\c'-z')\
dz.
By substituting the known value of the definite integral,
we have
The area of the generating circle is 7tc^, and the circumfer-
ence of the circle described by its centre is SttS. The product
of these two quantities is 'HiTt^hc'. Hence:
The volume of a circular ring is equal to the product of
the area of its cross-section into the circumference of its central
circle.
EXAMPLES AND EXERCISES.
1. Compare the cycloid with the semi-ellipse having the
same axes as the cycloid, and show the following relations be-
tween them:
a. The maximum radius of curvature of the ellipse (at the
point B) is greater than that of the cycloid in the ratio
Ti" : 2i, ox b : 4, nearly.
13. The area of the semi-ellipse is greater than that of the
cycloid in the ratio tt : 3.
y. The volume of the ellipsoid of revolution around the
axis OX is greater than that generated by the revolution of
the cycloid in the ratio 16 : 15.
30
306
TEE INTEGRAL CALCULUS.
n
K
Fig. 03.
199. Quadrature of Surfaces of
Revolution. Let us put q^
.^s = a small arc PQ oi a, curve re-
volving round an axis OX;
y = the distance of F from the '
axis OX;
y' = the distance of Q from the p'v
axis OX.
Considering Js as a straight line,
the surface generated by it will be the curved surface of the
frustum of a cone. If we put
/1(T = the area of this curved surface, we have, by Geometry,
A(T = 7t{y + y')^s.
Now let As become infinitesimal. Then y' will approach y
as its limit, and we shall have, for the differential of the sur-
face.
d(T = 'Hityds ■■
2^2/[l +
This expression, when integrated between the limits a;„ and
a;,, will give the area of that portion of the surface for which
the co-ordinates x are contained between «;„ and a;,.
The modifications and transformations of this formula so
as to apply it to cases when another axis than that of Y is
the axis of revolution, or when the equation of the curve is
not in the form y = <p{x), can be made by the student himself.
300. Examples of Surfaces of Revolution. The process
of applying the general formula for da- to special cases is so
like that already followed in quadrature and cubature that
the briefest indications will suffice to guide the student.
Surface of the Sphere. Supposing the equation of the gen-
erating circle to be written in the form
x'^y' = a\
SUBFA0E8 OF REVOLUTION. 307
we shall find the differential of the surface to be
d<T = %7tadx.
Prom this we may easily prove the following :
Theorem I, If a sphere be cut by any number of parallel
and equidistant planes, the curved surfaces of the spherical
zones contained between the planes will all be equal to each
other.
Theoeem II. Tlie total surface of a sphere is equal to the
product of its diameter and circumference.
Surf ace generated by the Revolution of a Cycloid. We shall
find the differential of the surface to be
d(y = ^Tta' sin' ^du.
By a formula found in Trigonometry, we have
8 sin' V = 6 sin w — 3 sin 3w.
Hence, putting v = ^m,
d<r — ^ltco' (3 sin v — sin dv)dv.
The whole surface is obtained by integrating between the
limits M = and ?< = 27r; that is, v = and v — it. We
thus find, for the total surface,
a = s^Tta'.
Hence the theorem:
The total surface generated by the revolution of a cycloid
about its base is four thirds the surface of the greatest in'
scribed sphere.
The Paraboloid of Revolution. Taking the integral be-
tween the limits zero and x, we have for the curved surface
THE END,
Cornell Uttfemitg Jitotg
THE ©I FT OF
3w|.J?,,C., Ca^I«/.
4.VJI1I : tf/a/tf
y ^ ^
'i^t jK'ff/' t
'y /-/' if •
